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A COURSE IN
MATHEMATICAL ANALYSIS
BY
6D0UARD GOURSAT
"PBOwnaaoH op Mathematics in the Univeiwity ok Paris
TRANSLATED BY
EARLE RAYMOND HEDRICK
Pbofessob op Mathematics in the University ok Missouri
Vol. I
DERIVATIVES AND DIFFERENTIALS
DEFINITE INTEGRALS EXPANSION IN SERIES
APPLICATIONS TO GEOMETRY
GINN AND COMPANY
BOSTON • NKW YORK • CHICAOO • LONDON
ATI AM A DALLAS • COLl'MBUS • SAN FHANCISCO
3)o3
V.J
ENTEBBD AT STATIONERS* HALL
COPTBIGHT, 1904, BY
EARLE RAYMOND HEDRICK
ALL RIGHTS RESERVED
PBIKTED IN THE UNITED STATES OF AMERICA
436.4
CINN ANU COMPANY • PRO-
PBIBTOM . MtTON • MAJk.
AUTHOR'S PREFACE
This book contains, with slight variations, the material given in
my course at the University of Paris. I have modified somewhat
the order fullowed in the lectures for the sake of uniting in a single
volume all that has to do with functions of real variables, except
the theory of differential equations. The differential notation not
being treated in the ** Classe de Math^matiques sp^iales," ♦ I have
treated this notation from the beginning, and have presupposed only
a knowledge of the formal rules for calculating derivatives.
Since mathematical analysis is essentially the science of the con-
tinuum, it would seem that every course in analysis should begin,
logically, with the study of irrational numbers. I have supposed,
however, that the student is already familiar with that subject. The
theory of incoiiimensurable numbers is treated in so many excellent
well-known works f that I have thought it useless to enter upon such
a discussion. As for the other fundamental notions which lie at the
basis of analysis, — such as the upper limit, the definite integral, the
double integral, etc., — I have endeavored to treat them with all
desirable rigor, seeking to retain the elementary character of the
work, and to avoid generalizations which would be superfluous in a
book intended for purposes of instruction.
Certain paragraphs which are printed in smaller type than the
body of the book contain either problems solved in detail or else
• An iDteresting account of French methods of instruction in mathematics will
be found in an article by Pierpont, BuUetin Amer. Math. Society, Vol. VI, 2d series
(1900), p. 225.— Trans.
t Such books are not common in En^jlish. The reader is referred to Pierpont,
Theory of Functions of Real I'ariahlefi, Ginn & Comimny, Boston. 1905; Tanner>*,
Lerofus d'arithmHique, 1900, and other foreign works on arithmetic and on real
functions.
ill
iT AUTHOR'S PREFACE
supplementary matter which the reader may omit at the first read-
ing without inconvenience. Each chapter is followed by a list of
exjunplei which are directly illustrative of the methods treated in
the chapter. Most of these examples have been set in examina-
tions. Certain others, which are designated by an asterisk, are
•omewhat more difficult The latter are taken, for the most part,
from original memoirs to which references are made.
Two of my old students at the ^^cole Normale, M. ^^mile Cotton
and }L Jean Clairin, have kindly assisted in the correction of proofs ;
I take this occasion to tender them my hearty thanks.
J«o«,87.1»(« B.GOUBSAT
TRANSLATOR'S PREFACE
The translation of this Course was undertaken at the suggestion
of Professor W. F. Osgood, whose review of the original appeared
in the July number of the Bulletin of the American Mathematical
Society in 1903. The lack of standard texts on mathematical sub-
jects in the English language is too well known to require insistence.
I earnestly hope that this book will help to fill the need so generally
felt throughout the American mathematical world. It may be used
conveniently in our system of instruction as a text for a second course
in calculus, and as a book of reference it will be found valuable to
an American student throughout his work.
Few alterations have been made from the French text. Slight
changes of notation have been introduced occasionally for conven-
ience, and several changes and additions have been made at the sug-
gestion of Professor Goursat, who has very kindly interested himself
in the work of translation. To him is due all the additional matter
not to be found in the French text, except the footnotes which are
signed, and even these, though not of his initiative, were always
edited by him. I take this opportunity to express my gratitude to
the author for the permission to translate the work and for the
sympathetic attitude which he has consistently assumed. I am also
indebted to Professor Osgood for counsel as the work progressed
and for aid in doubtful matters pertaining to the translation.
The publishers, Messrs. Ginn «& Company, have spared no pains to
make the typography excellent. Their spirit has been far from com-
mercial in the whole enterprise, and it is their hope, as it is mine,
that the publication of this book will contribute to the advance of
mathematics in America. ^ ^ HEDRICK
August, 1004
CONTENTS
Chaitbr Paob
I. Derivatives and Differentials 1
I. Functions of a Single Variable 1
II. Functions of Several Variables . . .11
III. The Differential Notation 10
II. Implicit Functions. Functional Determinants. Change
OF Variable 35
I. Implicit Functions 35
II. Functional Determinants 52
III. Transformations ... .... 61
III. Taylor's Series. Elementary Applications. Maxima
and Minima
I. Taylor's Series with a Remainder. Taylor's Series
II. Singular Points. Maxima and Minima .
IV. Definite Integrals
I. Special Methods of Quadrature ....
II. Definite Integrals. Allied Geometrical Concepts .
III. Change of Variable. Integration by Parts
IV. Generalizations of the Idea of an Integral. Improper
Integrals. Line Integrals
V. Functions defined by Definite Integrals .
VI. Approximate Evaluation of Definite Integrals
V. Indefinite Integrals
I. Integration of Rational Functions ....
II. Elliptic and Ilyperelliptic Integrals
III. Integration of Transcendental Functions
VL Double Integrals
I. Double Integrals. Methinls of Evaluation. (Jr»'»'U*s
Theorem . .
n. Change of Variable.s. An-a oi a Miria««'
III. Generalization.s of Double Integrals. Improper Int<»grals
Surface Integrals ......
IV. Analytical and Geometrical Applications
vil
80
110
134
134
140
166
175
192
106
208
208
L>>6
236
250
250
264
284
Yiii CONTENTS
Paox
VIL MVLTIPLB IlTTBORALS. IhTEORATION OF TOTAL DlFFER-
EKTIALS 296
I. Multiple IntegralB. Change of Variables .296
n. iBtagration of Totel Differentials 813
niL IxrufiTK SsRiES 327
L Series of Real Constant Terms. General Properties.
Tests for Convergence . . . 327
n. Series of Complex Terms. Multiple Series 350
HL Series of Variable Terms. Uniform Convergence . . 860
IX. POWRR SkRIES. TRIOOirOMETRFC SERIES .... 875
I. Power Series of a Single Variable 875
n. Power Series in Several Variables 894
in. Implicit Functions. Analytic Curves and Surfaces . 399
IV. Trigonometric Series. Miscellaneous Series . . .411
X. Plane Curves 426
I. Envelopes 426
n. Curvature 433
ni. Contact of Plane Curves 448
XI. Skew CrRVEs 453
I. Osculating Plane 453
II. Envelopes of Surfaces 459
III. Curvature and Torsion of Skew Curves .... 468
IV. Contact between Skew Curves. Contact between Curves
and Surfaces 486
Xn. BuwwAcms 497
I. Curvature of Curves drawn on a Surface . 497
n. Asjrmptotic Lines. Conjugate Lines .... 506
in. Lines of Curvature 514
IV. Families of Straight Lines 526
'>"»»» 541
I
A COURSE m MATHEMATICAL
ANALYSIS
CHAPTER I
DERIVATIVES AND DIFFERENTIALS
L FUNCTIONS OF A SINGLE VARIABLE
1. Limits. When the successive vahies of a variable x approach
nearer and nearer a constant quantity a, in such a way that the
absolute value of the difference x — a finally becomes and remains
less than any preassigned number, the constant a is called the
limit of the variable x. This definition furnishes a criterion for
determining whether a is the limit of the variable x. The neces-
sary and sufficient condition that it should be, is that, given any
positive number «, no matter how small, the absolute value of x — a
should remain less than < for all values which the variable x can
assume, after a certain instant.
Numerous examples of limits are to be found in Geometry
and Algebra. For example, the limit of the variable quantity
05 = (a' — m^) / (a — /w), as m approaches a, is 2 a ; for x — 2a will
be less than e whenever m — a is taken less than e. Likewise, the
variable x = a — 1 /n, where w is a positive integer, approaches the
limit a when n increases indefinitely ; for a — a* is less than c when-
ever n is greater than l/«. It is apparent from these examples that
the successive values of the variable x, as it approaches its limit, may
form a continuous or a discontinuous sequence.
It is in general very difficult to determine the limit of a variable
quantity. The following proposition, which we will assume as self-
evident, enables us, in many cases, to establish the existence of a limit
Any variable quantity which nether decreases, and which always
remains less than a constant quantity L, approaches a limit I, which
is less than or at most equal to L.
Similarly, any variable quantity which never increases^ and whieh
always remains greater than a constant quantity L\ approaches a
limit I'f which is greater than or else equal to L\
1
S DERIVATIVES AND DIFFERENTIALS [I, §2
''For example, if each of an infinite series of positive terms is
leM, wspeotiTely, than the corresponding term of another infinite
•eriM of positiYe terms which is known to converge, then the first
mrim oooTerges also ; for the sum 2, of the first n terms evidently
incieasM with n, and this sum is constantly less than the total sum
S of the second series.
2. FuncUont. When two variable quantities are so related that
the ralue of one of them depends upon the value of the other, they
are said to be functions of each other. If one of them be sup-
posed to vary arbitrarily, it is called the independent variable. Let
this rariable be denoted by a:, and let us suppose, for example,
that it can assume all values between two given numbers a and b
(a < b). Let y be another variable, such that to each value of x
b e t w e en a and 6, and also for the values a and b themselves, there
oorresponds one definitely determined value of y. Then y is called
a function of ar, defined in the interval (a, b) ; and this dependence
is indicated by writing the equation y —f(x). For instance, it may
happen that y is the result of certain arithmetical operations per-
formed upon X. Such is the case for the very simplest fimctions
studied in elementary mathematics, e.g. polynomials, rational func-
tions, radicals, etc.
A function may also be defined graphically. Let two coordinate
axes OXf Oy be taken in a plane ; and let us join any two points A
and B of this plane by a curvilinear arc A CB, of any shape, which
is not cut in more than one point by any parallel to the axis Oy.
Then the ordinate of a point of this curve will be a function of the
a bsc i ssa. The arc A CB may be composed of several distinct por-
tions which belong to'&ifferent curves, such as segments of straight
lines, arcs of circles, etc.
In short, any absolutely arbitrary law may be assumed for finding
the ralue of y from that of x. The word function, in its most gen-
tial sense, means nothing more nor less than this : to every value of
s eonesponds a value of y.
8. Continnity. The definition of functions to which the infini-
tesimal oaloulus applies does not admit of such broad generality.
Lai y -/(as) be a function d%gned in a certain interval (a, 6), and
let «, and «t + A be two values of x in that interval. If the differ-
/('• + *)-/(«o) approaches zero as the absolute value of h
sro, the function /(x) is said to be continuous for the
m^ From the very definition of a limit we may also say that
1,53] FUNCTIONS OF A SINGLE VARIABLE 8
a function f{x) is continuous for x = Xo if, corresponding to every
positive number «, no matter how smcUly we can find a positive num^
ber rii such that
|/(x. + A)-/(x„)|<.
for every value of h less than rj in absolute value.* We shall say that
a function f(x) is continuous in an interval (a, b) if it is continuous
for every value of x lying in that interval, and if the differences
/(a + A)-/(a), fib -h)- fib)
each approach zero when A, which is now to be taken only positive,
approaches zero.
Irt elementary text-books it is usually shown that polynomials,
rational functions, the exponential and the logarithmic function,
the trigonometric functions, and the inverse trigonometric functions
are continuous functions, except for certain particular values of
the variable. It follows directly from the definition of continuity
that the sum or the product of any number of continuous functions
is itself a continuous function ; and this holds for the quotient of
two continuous functions also, except for the values of the variable
for which the denominator vanishes.
It seems superfluous to explain here the reasons which lead us to
assume that functions which are defined by physical conditions are,
at least in general, continuous.
Among the properties of continuous functions we shall now state
only the two following, which one might be tempted to think were
self-evident, but which really amount to actual theorems, of which
rigorous demonstrations will be given later, f
I. If the function y =f(x) is continuous in the interval (a, ft), and
if N is a number between f (a) a7idfib)y then the equation f(x) = N
has at least one root between a and b.
II. There exists at least one value of x belonging to the interval
(a, b), inclusii^e of its end points^ for which y takes on a value M
which is greater than, or at least equal to, the value of the function at
any other point in the interval. Likewise, there exists a value of x
for which y takes on a value m, than which the function assumes no
smaller value in the interval.
The numbers M and m are called the maximum and the minimum
values of f(x), respectively, in the interval (o, h). It is clear th^
* The notation \ a | denotes the absolute value of a.
t See Chapter IV.
4 I>BRIVATrVES AND DIFFERENTIALS [I, §4
the ralue of x tat which /(x) assumes its maximum value AT, or the
ralne of x oonesponding to the minimum m, may be at one of the
end points, a or 6. It follows at once from the two theorems above,
that if iV is a number between M and m, the equation f{x) = N has
at least one root which lies between a and h.
4L Examples of discontinuities. The functions which we shall study
will be in general continuous, but they may cease to be so for
certain exceptional values of the variable. We proceed to give
several examples of the kinds of discontinuity which occur most
frequently.
The function y = l/(x — a) is continuous for every value Xq of
X except a. The operation necessary to determine the value of y
frt>m that of X ceases to have a meaning when x is assigned the
value a ; but we note that when x is very near to a the absolute
value of y is very large, and y is positive or negative with x — a.
Ab the difference x — a diminishes, the absolute value of y increases
indefinitely, so as eventually to become and remain greater than any
praasaigned number. This phenomenon is described by saying that
jf beeomes infinite when x = a. Discontinuity of this kind is of
great importance in Analysis.
Let us consider next the function y = sin 1 /x. As x approaches
xero, 1 /x increases indefinitely, and y does not approach any limit
whatever, although it remains between + 1 and - 1. The equation
nnl/x = Ay where|^|<l, has an infinite number of solutions
which lie between and c, no matter how small e be taken. What-
ever value be assigned to y when a; = 0, the function under con-
sideration cannot be made continuous for x = 0.
An example of a still different kind of discontinuity is given by
the convergent infinite series
"(^> = ^"^iT^-^ • +
(1 + xy
When • approaches zero, S(x) approaches the limit 1, although
5(0) mm 0. For, when a: = 0, every term of the series is zero, and
heooo S(0) = 0. But if a; be given a value different from zero, a
geometric progression is obtained, of which the ratio is 1/(1 -|- x^
8(x) f!_ = fl(l±f3-i .,..
I.§a] FLT^CTIONS OF A SINGLE VARIABLE 5
and the limit of S(x) is seen to be 1. Thus, in this example, the
function approaches a delinite limit as x approaches zero, but that
limit is different from the value of the function for x = 0.
5. Derivatives. Let/(a:) be a continuous function. Then the two
terms of the quotient
/(x + A)-/(ar)
A
approach zero simultaneously, as the absolute value of h approaches
zero, while x remains fixed. If this quotient approaches a limit,
this limit is called the derivative of the function /(a*), and is denoted
by I/', or by /' (x), in the notation due to Lagrange.
An important geometrical concept is associated with this analytic
notion of derivative. Let us consider, in a plane XOY, the curve
A MB, which represents the function y =/(x), which we shall assume
to be continuous in the interval (a, b). Let M and M' be two points
' on this curve, in the interval (a, b), and let their abscissai be x and
X 4- A, respectively. The slope of the straight line MM' is then
precisely the quotient above. Now as h approaches zero the point
M' approaches the point 3/; and, if the function has a derivative,
the slo|)e of the line MM' approaches the limit y'. The straight line
MM', therefore, approaches a limiting position, which is called the
tangent to the curve. It follows that the equation of the tangent is
Y-y = y'{X-x),
where X and Y are the running coordinates.
To generalize, let us consider any curve in space, and let
be the coordinates of a point on the curve, expressed as functions of
a variable parameter t. Let M and M' be two points of the curve
corresponding to two values, t and t -\- h, of the parameter. The
equations of the chord MM' are then
X-f(f) ^ Y-<t>(f) ^ Z-^(t)
f{t + h) -f(t) <i>{t + h) - <l>{t) ^(t 4- h) - ^(t)
If we divide each denominator by h and then let h approach zero,
the chord Myf evidently approaches a limiting position, which is
given by the equations
fit) <t>'(t) ^'(t) '
6 DERIVATIVES AND DIFFERENTIALS [I, §5
proTided, of course, that each of the three functions /(O, <^(0» ^(0
potMtaes a derivative. The determination of the tangent to a curve
thus wducea, analytically, to the calculation of derivatives.
Every function which powesses a derivative is necessarily con-
tinuous, but the converse is not true. It is easy to give examples
of continuous functions which do not possess derivatives for par-
ticular values of the variable. The function y = xsinl/a;, for
example, is a perfectly continuous function of a, for x = 0,* and y
approacherzero as x approaches zero. But the ratio y jx = sinl/o;
does not approach any limit whatever, as we have already seen.
Let us next consider the function y = xl Here y is continuous
for every value of a; ; and y = when a; = 0. But the ratio y/x=^x~^
increases indefinitely as x approaches zero. For abbreviation the
derivative is said to be infinite for x = ; the curve which repre-
sents the function is tangent to the axis of y at the origin.
Finally, the function
xe^
y = i
is continuous at x = 0,* but the ratio y /x approaches two different
limits according as a; is always positive or always negative while
it is approaching zero. When x is positive and small, e^/"" is posi-
tive and very large, and the ratio y /x approaches 1. But if x
is negative and very small in absolute value, e^^^ is very small, and
the ratio y /x approaches zero. There exist then two values of the
derivative according to the manner in which x approaches zero : the
curve which represents this function has a comer at the origin.
It is clear from these examples that there exist continuous func-
tions which do not possess derivatives for particular values of the
variable. But the discoverers of the infinitesimal calculus confi-
dently believed that a continuous function had a derivative in gen-
eral. Attempts at proof were even made, but these were, of course,
fallacious. Finally, Weierstrass succeeded in settling the question
conclusively by giving examples of continuous functions which do not
possess derivatives for any values of the variable whatever.f But
as these functions hseve not as yet been employed in any applications,
* AflUr the valne laro hat been auigned to y for x = C — Translator.
t Note tmd at the Academy of Science! of Berlin, July 18, 1872. Other examples
are to ba foaad la the memoir by Darboux on discoutinuous functions (Annales de
finU NcrmaU SupHimtn, Vol. IV, 'id series). One of Weierstrass's examples is
flvMi later (Chapter IX).
1,56] FUNCTIONS OF A SINGLE VARIABLE 7
we shall not consider them here. In the future, when we say that
a function f{x) has a derivative in the interval (a, 6), we shall mean
that it has an unique Jinite derivative for every value of x between
'/ jiiid b and also for x = a (A being positive) and for x = b (h being
negative), unless an explicit statement is made to the contrary.
6. Successive derivatives. The derivative of a function f(x) is in
general another function of x,f'(x). U f'(x) in turn has a deriva-
tive, the new function is called the second derivative of /^, and is
represented by y" or by f"(x). In the same way the third deriva-
tive I/'", or f"'{x), is defined to be the derivative of the second, and
so on. In general, the nth derivative y<">, or /^"^(a;), is the deriva-
tive of the derivative of order (n — 1). If, in thus forming the
successive derivatives, we never obtain a function which has no
derivative, we may imagine the process carried on indefinitely. In
this way we obtain an unlimited sequence of derivatives of the func-
tion f(x) with which we started. Such is the case for all functions
which have found any considerable application up to the present
time.
The above notation is due to Lagrange. The notation /)^y, or
D^f(x)f due to Cauchy, is also used occasionally to represent the
7ith derivative. Leibniz' notation will be given presently.
7. RoUe's theorem. The use of derivatives in the study of equa-
tions depends upon the following proposition, which is known as
RoUe's Theorem:
Let a and b be two roots of the equation f{x)= 0. If the function
f(x) is continuous and possesses a derivative in the irUerval {ay ft),
the equation f'(x) = has at least one root which lies between aandb.
For the function f{x) vanishes, by hypothesis, for x = a and x = b.
If it vanishes at every point of the interval (a, i), its derivative also
vanishes at every point of the interval, and the theorem is evidently
fulfilled. If the function /(a;) does not vanish throughout the inter-
val, it will assume either positive or negative values at some points.
Suppose, for instance, that it has positive values. Then it will have
a maximum value M for some value of x, say Xj, which lies betweei:
a and b (§ 3, Theorem II). The ratio
f{x, + h)-f{x,)
h '
DKBIVATIVES AND DIFFERENTIALS [I. §8
k is taken positive, is necessarily negative or else zero.
Uia limit of this ratio, i.e. /'(xi), cannot be positive; i.e.
/*(«!) S®- ^^ ^ ^® consider /'(a:i) as the limit of the ratio
/(x,-h)-f(x,)
— h
where A U poeitive, it follows in the same manner that f'(xi) > 0.
two results it is evident that/'(xi) = 0.
8. Lew of the mean. It is now easy to deduce from the above
theorem the important law of the mean:'**'
Lei f{x) be a eontinuotu function which has a derivative in the
imierval (a, 6). Then
a) m-f{a) = (h-a)f(c),
where e %» a number between a and h.
In order to prove this formula, let <^ (x) be another function which
hea the same properties as f{x)) i.e. it is continuous and possesses a
derivative in the interval (a, b). Let us determine three constants,
A,BfCj such that the auxiliary function
^(x)=^Af(x) + B<f>(x)-\-C
TMiishes for x = a and for x = b. The necessary and sufficient
conditions for this are
i4/(a)+ 5^(«)+ C = 0, Af(b) + B<t»(b)+ C = 0;
and these are satisfied if we set
^ =^(a)-^.(*)» ^=/W-/(a), C=f(a)<l>(b)-f(b)<f>(ay
The new function ^(x) thus defined is continuous and has a derivative
In the interval (a, b). The derivative i/^'(ar) = A f'(x) + B <f>'(x) there-
fore ranishes for some value c which lies between a and b, whence,
replaoiag A and B by their values, we find a relation of the form
It is merely neceesary to take it»(x) = x in order to obtain the equality
which was to be proved. It is to be noticed that this demonstration
does not presuppose the continuity of the derivative /'(a;).
.. The French also use "Formule de la
My«MM M s qrBMym. Othar BngUah lynonymi are •• Average value theorem "
••e lUM valiM iheowD." — Tkani.
I, §8] FUNCTIONS OF A SINGLE VARIABLE 9
From the tiieoieiii just proven it follows tliat it the derivative
f'(x) is zero at each point of the interval (a, b), the function f{x)
has the same value at every point of the interval ; for the applicar
tion of the formula to two values Xiy x,, belonging to the interval
(a, b)f gives /(x,)=/(xj). Hence, if two functions have the same
derivative, their difference is a constant; and the converse is evi-
dently true also. If a function F(x) be given whose derivative is
f{x), all other functions which have the same derivative are found by
adding to F{x) an arbitrary cotistant.*
The geometrical interpretation of the equation (1) is very simple.
Let us draw the curve A MB which represents the function g =f(^x)
in the interval (a, b). Then the ratio [/(*) —/(«)]/(* — «) is the
slope of the chord AB, while f'(c) is the slope of the tangent at a
point C of the curve whose abscissa is c. Hence the equation (1)
expresses the fact that there exists a point C on the curve A MB,
between A and B, where the tangent is parallel to the chord AB.
If the derivative f\x) is continuous, and if we let a and b approach
the same limit Xq according to any law whatever, the number c,
which lies between a and b, also approaches Xo> a-nd the equation (1)
shows that the limit of the ratio
m-m
b — a
is /'(^o)- The geometrical interpretation is as follows. Let us
consider upon the curve y —f{x) a point M whose abscissa is Xq,
and two points A and B whose abscissae are a and b, respectively.
The ratio [f(b)—f{a)']/{b — a) is equal to the slope of the chord
AB, while f'(Xo) is the slope of the tangent at ^f. Hence, when
the two points .4 and B approach the point M according to any law
whatever, the secant yl 5 approaches, as its limiting position, the
tangent at the point M.
• This theorem is sometimes applied without due regard to the conditions imposed in
its statement. I^t/(x) and <p{x), for example, be two continuous functions which have
derivatives /'(x), <f>'(x) in an interval (a, 6). If the relation /'(ar) <p{x)-/{r) 4>'{x) =
is satisfie*! by these four functions, it is sometimes accepted as proved that the deriva-
tive of the function// 0, or [/'(x) <t> (x) - f{x) 4>'{x)] / ^«, is zero, and that accordingly
//0 is constant in the interval (a, 6). But this conclusion is not absolutely rigorous
unless tlje function {x) does not vanish in the interval (a, 6). SupiKwe, for instance,
that {x) and 4>'(x) b»th vanish for a value c between a and b. A function /(jr) equal
to Cx4>{x) between a and c, and to C^tpix) In'twcen c and 6. where C^ and Cj are dif-
ferent constants, is continuous and has a derivative in the interval (a, 6), and we have
f'(x) <p (x) -f{x) (p'(x) - for every value of x in the interval. The geometricftl
Interpretation is apparent.
10
DERIVATIVES AND DIFFERENTIALS
[I, §9
This does not hold in general , however, if the derivative is not
ooDtuiuous. For instance, if two points be taken on the curve
y s x', on opposite sides of the y axis, it is evident from a figure
that the direction of the secant joining them can be made to approach
any arbitrarily assigned limiting value by causing the two points to
approach the origin according to a suitably chosen law.
The eqtiation (T) is sometimes called the generalized law of the
SMon. From it de I'Hospital's theorem on indeterminate forms fol-
lows at once. For, suppose f{a) — and ^ (a) = 0. Replacing h
by Jr in (lOf we find
where j'j lies between a and x. This equation shows that if the
ratio f'(x)/<^'(x) approaches a limit as x approaches a, the ratio
f{x)/^{x) approaches the same limit j if f(a) = and <f>(a) = 0.
9. Generalizations of the law of the mean. Various generalizations of the law
of the mean have been suggested. The following one is due to Stieltjes {Bulletin
de la SotAiti Math^matigue, Vol. XVI, p. 100). For the sake of definiteness con-
■ider three functions, /(x), g{x), h{x), each of which has derivatives of the first
and aecond orders. I^et a, b, c be three particular values of the variable (a < 6 < c).
Lei .^ be a number defined by the equation
and let
*(«) =
/(a) g{a) h{a)
f{h) g{b) h{b)
/(c) g{c) h{c)
f{o) g{a) h{a)
/(&) g{b) h{b)
fix) 9{x) h{x)
1 a a2
-A
1 6 62
1 c c2
1 a a2
-A
1 6 62
1 X X2
0,
be aa auxiliary function. Shice this function vanishes when a; = 6 and when
X = c, iu derivaUve must vanish for some value f between 6 and c. Hence
/(a) 9 {a) h{a)
/(&) 9{b) h{b)
rU) fl^(f) A'(i-)
-A
1 a o2
1 6 62
1 2t
= 0.
If 6 be replaeed by z In the left-hand side of this equation, we obtain a function
rll******* ▼antahea when x = a and when x = 6. Its derivative therefore van-
lahee for some value of x between a and 6, which we shall call {. The new
•qaatlon thus obcained it
/(a)
/'(O
fV)
9 (a)
9'(C)
h{a)
hV)
-A
a2
2r
= 0.
FtoaMy, repUelng f hy x In the lefuhand side of this equation, we obtain a func-
lion of X which vanlabee when x = e and when x = f . Its derivative vanishes
I, §10] FUNCTIONS OF SEVEUAL VARIABLES 11
for some value 17, which lien between ( and ^ and therefore between a and c
Hence A muBt have the value
-n
/(a) g{a) h (a)
r{i) 9'{i) h'{k)
riyi) ff"(v) fi"iri)
where { lies between a and 6, and »j lies between a and c.
This proof does not presuppone the continuity of the second deriratbei
/"(x), g"{x), h"{x). If these derivatives are continuous, ami if the values a^b^e
approach tlie same limit Xq, we have, in the limit,
lim^=i
/ (Xo) g (Xo) h (Xo)
r{Xo) g'(Xo) A'(xo)
/"(Xo) g^'ixo) /i"(xo)
Analogous expressions exist for n functions and the proof follows the same
lines. If only two functions /(x) and g (x) are taken, the formulae reduce to the
law of the mean if we set g{x) = I.
An analogous generalization has been given by Schwarz (Annali di Math&-
matica, 2d series, Vol. X).
II. FUNCTIONS OF SEVERAL VARIABLES
10. Introduction. A variable quantity <i> whose value depends on
the values of several other variables, x, y, z, •••, t, which are in-
dependent of each other, is called a function of the independ-
ent variables x, y, «, •••, t; and this relation is denoted by writing
o) =/(a;, i/,z,--y t). For definiteness, let us suppose that o> =f(x, y)
is a function of the two independent variables x and y. If we think
of X and y as the Cartesian coordinates of a point in the plane,
each pair of values (x, y) determines a point of the plane, and con-
versely. If to each point of a certain region A in the xy plane,
bounded by one or more contours of any form whatever, there
corresponds a value of w, the function /(x, y) is said to be defined
in the region A.
Let (xo, yo) be the coordinates of a point Mq lying in this r» .
The function f{x, y) is said to be continuous for the pair of r
(*o» yo) ifi corresponding to any preassigned positive number c, another
positive number tf exists such that
|/(xo -f A, yo 4- k) — /(Xo, yo)| < <
whenever \h\< rj and \k\< rj.
This definition of continuity may be interpreted as follows. Let
us suppose constructed in the xy plane a square of side 2i; about
3/0 as center, with its sides parallel to the axes. The point AT.
12 DERIVATIVES AND DIFFERENTIALS [I, §11
whoM oooidinates are »o 4- *, yo + *, will lie inside this square, if
|A|< If and |A| < ij. To say that the function is continuous for the
pair of Tidues (*•, yo) amounts to saying that by taking this square
■affioieoUy smaU we can make the difference between the value of
the function at M^ and its value at any other point of the square less
than c in absolute value.
It is evident that we may replace the square by a circle about
(x^ y,) as center. For, if the above condition is satisfied for all
pointo inside a square, it will evidently be satisfied for all points
inside the inscribed circle. And, conversely, if the condition is
satisfied for all points inside a circle, it will also be satisfied for all
points inside the square inscribed in that circle. We might then
define conti nuity by saying that an rj exists for every c, such that
whenever VA* + k* < 17 we also have
\f(xo -h A, yo + k) -f(xo, yo) | < c
The definition of continuity for a function of 3, 4, • • • , n inde-
pendent variables is similar to the above.
It is clear that any continuous function of the two independent
Tariables x and y is a continuous function of each of the vai-iables
taken separately. However, the converse does not always hold.*
IL Partial derivatives. If any constant value whatever be substi-
tuted for y, for example, in a continuous function /(ic, y), there
results a continuous function of the single variable x. The deriva-
tive of this function of x, if it exists, is denoted by f^(x, y) or by a>^.
Likewise the symbol w^, or /^ (x, y), is used to denote the derivative
of the function /(oj, y) when x is regarded as constant and y as the
independent variable. The functions f^ (x, y) and f^ (x, y) are called
the partial derivatives of the function /(x, y). They are themselves,
in general, functions of the two variables x and y. If we form their
partial derivatives in turn, we get the partial derivatives of the sec-
ond order of the given function /(x, y). Thus there are four partial
deriTatives of the second order,/^(x, y),/^(x, y),/,^(x, y), /^(.x, y).
The partial derivatives of the third, fourth, and higher orders are
ft for IBftaBM, the function /(z, y), which is equal to 2 xy / (x^ + y^ when
I two TsHablM c and y ar« not hoth zero, and which is zero when x — y = 0. It is
It It ■ ooDtinuoua function of x when y is constant, and trice versa.
MttMtailMM h It not a cootinuoua function of the two independent variables x and y
lor tiM pair of vahMt s s 0, y = 0. For, if the point (x, y) approaches the origin upon
Ikt llao a V y, tbt fnaoUoo/Cz, y) approached the limit 1 , and not zero. Such functions
tevo ktoa aladlod by Batra la hit thetli.
I, §iij FUNCTIONS OF SEVERAL VARIABLES 13
defined similarly. In general, given a function a> =/(ar, y, «, ••, ^)
of any number of independent variables, a partial derivative of the
nth order is the result of n successive differentiations of the function
/, in a certain order, with respect to any of the variables which occur
in/. We will now show that the result doesrfiot depend upon the
order in which the differentiations are carried out.
Let us first prove the following lenA|a :
Let o> =f(x, y) be a function of the two variqhles^Land y. Then
fxy — fyxi provided that these two derivatives are contu
To prove this let us first write the expression
U =/(x 4- Ax, y + Ay) -f(x, y -f Ay) -f(x -f Ax, y) hV(x, y)
in two different forms, where we suppose that x, y, Ax, ^^ have
definite values. Let us introduce the auxiliary function
<^00=/(« + Ax, i;)-/(x, v),
where i^ is an auxiliary variable. Then we may write
u = <f>(y + ^y)-<f>(y)-
Applying the law of the mean to the function <t>{v)f we have
U = Ay<t>,(y + ^Ay), where < d < 1 ;
or, replacing <f>^ by its value,
U = Ay[/,(x -h Ax, y 4- ^Ay) -/,(x, y -f ^Ay)].
If we now apply the law of the mean to the function /^ (m, y + dAy),
regarding u as the independent variable, we find
f/ = Ax Ay/^^(x -f d' Ax, y + ^Ay), < ^' < 1.
From the symmetry of the expression U in x, y. Ax, Ay, we see that
we would also have, interchanging x and y,
f/ = Ay Ax/^ (X + $[ Ax, y + $^ Ay),
where ^, and 0[ are again positive constants less than unity. Equat-
ing these two values of U and dividing by Ax Ay, we have
/^(x -I- ^, Ax, y 4- d, Ay) =^(x 4- ^'Ax, y 4- ^Ay).
Since the derivatives /^(x, y) and/^(x, y) are supposed continuous,
the two members of the above equation approach f^ (x, y) and
fwxi^f y)> respectively, as Ax and Ay approach zero, and we obtain
the theorem which we wished to prove.
14 DERIVATIVES AND DIFFERENTIALS [I, §11
It is to be noticed in the above demonstration that no hypothesis
whatever is made concerning the other derivatives of the second order,
/^ and /^ The proof applies also to the case where the function
/(x, y) depends upon any number of other independent variables
besides x and y, since these other variables would merely have to
be regarded as constants in the preceding developments.
Let us now consider a function of any number of independent
▼enables, M ^ ^, ..
and let O be a partial derivative of order n of this function. Ajiy
permutation in the order of the differentiations which leads to O
can be eCFected by a series of interchanges between two successive
differentlbttions ; and, since these interchanges do not alter the
resul^is we have just seen, the same will be true of the permuta-
tioi^Hsidered. It follows that in order to have a notation which
is dw ambiguous for the partial derivatives of the wth order, it is
sufficient to indicate the number of differentiations performed with
respect to each of the independent variables. For instance, any wth
derivative of a function of three variables, w =/(aJ, y, «), will be
represented by one or the other of the notations
where p -\- q -{■ r = n.* Either of these notations represents the
result of differentiating / successively p times with respect to x,
q times with respect to y, and r times with respect to z, these oper-
ations being carried out in any order whatever. There are three
distinct derivatives of the first order, /^, /y, /^ ; six of the second
order, /^, /^, /^, /^, /^, /^ ; and so on.
In general, a function of p independent variables has just as many
distinct derivatives of order n as there are distinct terms in a homo-
geoeous polynomial of order niup independent variables ; that is,
(n + l)(n-f 2).(n-f-jg-l)
1.2....(^-2)(^-l) '
as is shown in the theory of combinations.
Praetieal ruUi. A certain number of practical rules for the cal-
oulatioD of derivatiyes are usually derived in elementary books on
•Tb» MUtton A,^,r(x, y, t) is OMd instead of the notation /j^J^r (x, y, z) for
■Inplleitj. Tbna the iM>Ution/^(x, y), used in place of /;;(«, y), is simpler and
•qoall J ekar. — TaAM s.
1, §11] FUNCTIONS OF SKVERAL VARIABLES Ifi
the (Calculus. A table of such rules is appended, the function and
its derivative being placed on the saiae line :
y = a'y y' = a' log a,
where the symbol log denotes the natural logarithm ;
y = logx,
^-i^
y = sin Xj
y' = cos a- ;
y = cos a?,
y' = — sin X ;
«# *M» Q 1*/% ain o^
u'- 1 :
y = arc oiu X|
^ iVnriT.'
y = arc tan ar,
«'- 1 ;
^-l+x»'
y = wv,
y = u'v 4- uv' ;
y =/(«),
y.=f(y)u,',
y =/(«. ", «>),
y. = n,f,-^v,f,^w,f^
The last two rules enable us to find the derivative of a function
of a function and that of a composite function if /,,/»»/» are con-
tinuous. Hence we can find the successive derivatives of the func-
tions studied in elementary mathematics, — polynomials, rational
and irrational functions, exponential and logarithmic functions,
trigonometric functions and their inverses, and the functions deriv-
able from all of these by combination.
For functions of several variables there exist cfitain formulae
analogous to the law of the mean. Let us consider, for definite-
ness, a function /(x, y) of the two independent variables x and y.
The difference /(x + ^, y 4- k) —f{x, y) may be written in the form
/(x 4- h, y 4- k) -/(x, y) = [/(x + ^, y -f k) -f(r, y -\- A-)]
+ [/(^, y4-A:)-/(^>y)]»
to each part of which we may apply the law of the mean. We
thus find
f(x -^h,y + k) -/(x, y) = hf,(x + eh,y-^k)-\- kf^(x, y 4- B'k),
where 6 and $* each lie between zero and unity.
This formula holds whether the derivatives/, and/^ are continu-
ous or not. If these derivatives are continuous, another formula,
Uki
16 DERIVATIVES AND DIFFERENTIALS [I, §12
timiUr to the above, but involving only one undetermined number
0^ maj be employed.* In order to derive this second formula, con-
sider the auxiliary function <ft(t) =f(x + ht, y -\- kt), where x, y, A,
and k have determinate values and t denotes an auxiliary variable.
Applying the law of the mean to this fjtnction, we find
^(1)-<^(0)=<^W, b<^<i.
Now ^{i) is a composite fundtion of t^ and its derivative <^'(^) is
equal to A/,(x + V, y 4- A;^ + ^fy{x 4- ht, y + kt)-, hence the pre-
ceding formula may be written in the form
f(x-k-h,y-{'k)'-f{x,y) = hf,{x-iteh,y + ek) + kf^{x + eh,y+ek),
13. Tangent plane to a surface. We have seen that the derivative
of a function of a single variable gives the tangent to a plane curve.
Similarly, the partial derivatives of a function of two variables occur
in the determination of the tangent plane to a surface. Let
(2) z = F(x, y)
be the equation of a surface 5, and suppose that the function F(x, y),
together with its first partial derivatives, is continuous at a point
(«o» yo) of the xy plane. Let Zq be the corresponding value of «,
and il/o (a^o, yo, «o) the corresponding point on the surface S. If
the equations
(3) x=f{t), y = <t>(t), z = tf;(t)
reprasent a curve C on the surface S through the point Mq, the
three functions /(<), <t>(t), ^(^), which we shall suppose continuous
and differentiable, must reduce to Xq, yo, Zq, respectively, for some
ralue t^ of the parameter t. The tangent to this curve at the point
J/« is given by the equations (§ 5)
^ f{to) <!>%) ^'(to) '
Since the curve C lies on the surface S, the equation ^(t)=F[f(t), <f>(t)'\
must hold for all values of ^; that is, this relation must be an identity
• A a o tlfr lonDolA may be obtained which involves only one undetermined number d,
Bad which holds even whan thederlvatives/e and/^ are discontinuous. For the applica-
Uoiiof th« Uw of the mean to the auxiliaiy function 0(0 =/(x + H y + *)+/(«, y + AO
^ ^a)-^(O) = 0'(^, 0<tf<l,
/(« + *, If + *) -/(«» y) = A/,(x + tfA, y + t) + k/y(z, y + ek), 0<d<\.
The op eiat loM performed, and hence the final formula, all hold provided the deriva-
HvM/^Md/ymerely exist at the pointo (x + A<. y + *), (x, y + AO, 0<<^1. — Trans.
I, § 13] FUNCTIONS OF SEVERAL VARIABLES 17
in t. Taking the derivative of the second member by the rule for
the derivative of a composite function, and setting t = <©, we have
(5) fC.) =/'(<•) ^'.. + *'W^v
We can now eliminate f{t^, ♦'('•)' ^'(M between the equations (4)
and (5), and the result of this elimination is
(6) Z-x, = {X-x,)F^+(^Y-y,)F,^.
This is the ecjuation of a plane which is the locus of the tangents to
all curves on the surface through the point Mq. It is called the tan-
gent plane to the surface,
18. Passage from increments to derivatives. We have defined the BUCcesBive
derivatives in terms of each other, the derivatives of order n being derived from
those of order (/i - 1), and so forth. It is natural to inquire whether we may
not define a derivative of any order as the limit of a certain ratio directly, with-
out the intervention of derivatives of lower order. We have already done some-
thing of this kind lor f^y (§ 11); for the demonstration given above shows that/a^
is the limit of the ratio
/(x -h Aa, y -f Ay) -/(x + Aa;, y) -/(g, y + Ay) -f /(g, y)
Ax Ay
as Ax and Ay both approach zero. It can be shown in like manner that the
second derivative /" of a function /(x) of a single variable is the limit of the
ratio
fix -t- Ai + ^) -/(x -i- ^0 -/(x 4- h^) -H/(x)
hxho
as h\ and hi both approach zero.
For, let us set
/i(«)=/(x + Ai)-/(x),
and then write the above ratio in the form
/i(x-f A»)-/i(x) ^ /{(x-f-gA,) o«^<l;
hxh^ hi
or
hi
The limit of this ratio is therefore the second derivative /", provided that
derivative is continuous.
Passing now to the general case, let us consider, for definiteness, a function of
three independent variables, w =/(x, y, z). Let us set
Ajw =/(x + A, y, z) -/(x, y, «),
aJw =/(x, y + *, «) -/(x, y, «),
a[« =/(x, y, z + -/(x, y, 2),
where aJ w, aJ w, A^ w are i\\Q first increnxents of w. If we consider A, Jr, < as givW)
constants, then the.se three first increments are themselves functions of x, y, s,
and we may form the relative increments of these functions corresponding to
18 DERIVATIVES AND DIFFERENTIALS [I, §13
tDcremenu Ai, i^i, h of the variables. This gives us the second increments,
^^^, A*i A* w, • • • . This process can be continued indefinitely ; an increment
of Older n would be defined as a first increment of an increment of order (n — 1).
8iiieo we may invert the order of any two of these operations, it will be suffi-
doDt to tndifiatt the successive increments given to each of the variables. An
increment of order n would be indicated by some such notation as the following :
AC)« = aJ« A^^ . . . Ai' aJ' . . . aJ» aJ . . . a';/{z, y, «),
p + q -^ r = n^ and where the increments h, k, I may be either equal or
This increment may be expressed in terms of a partial derivative of
order », being equal to the product
X /.f»i,r(x + $ihi + • • • + 0php, y + eikx + . . . + e'^kq, z 4- e^h + • • • + K'lr),
where every B lies between and 1. This formula has already been proved foi
first and for second increments. In order to prove it in general, let us assume
that it holds for an increment of order (n — 1), and let
0(2, y, z) = aJ« . . . aJ'' Ajt . . . aJ'A^ . . . aIv.
Then, by hypothesis,
#(Xt y, «) = ^- • • Ap*l • • Kh • • • lrfxP-l^zr{X-\-e2h^-\-' . . +^p^, y + . . ., Z+ . . .).
Bat the nth increment considered is equal to 0(x + ^i, y, z) — 0(x, y, z) ; and if we
apply the law of the mean to this increment, we finally obtain the formula sought.
Conversely, the partial derivative fxP^flz* is the limit of the ratio
A^AJi'-. ■ A^A^t. .. A^^A^ • • A,V
h\hi' hpkiki- • -kqli" Ir
M all the increments A, k, I approach zero.
It is interesting to notice that this definition is sometimes more general than
the Qsoal definition. Suppose, for example, that w =/(x, y) = <P{x) + yf/ (y) is a
function of x and y, where neither <f> nor ^ has a derivative. Then <a also has
no first derivative, and consequently second derivatives are out of the question.
In the ordinary sense. Nevertheless, if we adopt the new definition, the deriva-
UTe/ay is the limit of the fraction
fix -\.h,y-\-k) -fix + h, y) -fix, y + k) -\-f{z, y)
hk
irhich Is equal to
^(x -f ») -f f (y -t- ie) - 0(g -f. A) - ^(y) ~ »(x) -- ^(y + fe) -f ^(x) -\- ^(y)
hk
But the numerator of this ratio is identically zero. Hence the ratio approaches
wro as a limit, and we find /ry = 0.*
* ▲ ilmUar remark may be made regarding functions of a single variable. For
example, the ftuiotlon/(z) = z* cos 1/x has the derivative
X z
•J>d/'(«) has no derivative for a; = 0. But the ratio
/(2a)-2/(a)-f/(0)
-J
or SaeotO/So) — aaeoeO/or), has the limit sero when a approaches sero.
I, il4] THE DIFFERENTIAL NOTATION 19
in. THE DIFFERENTIAL NOTATION
The differential notation, which has been in use longer than any
other,* is due to Leibniz. Although it is by no means indispensable,
it possesses certain advantages of symmetry and of generality which
are convenient, especially in the study of functions of several varia-
bles. This notation is founded upon the use of infinitesimals.
14. DifferentUls. Any varuible quantity which approaches zero as
a limit is called an infinitely small quantity^ or simply an infinitesi-
mal. The condition that the quantity be variable is essential, for
a constant, however small, is not an infinitesimal unless it is zero.
Ordinarily several quantities are considered which approach zero
simultaneously. One of them is chosen as the standard of compari-
son, and is called the principal infinitesimal. Let a be the principal
infinitesimal, and ft another infinitesimal. Then $ is said to be an
infinitesimal of higher order ivith respect to a, if the ratio ft/ a
approaches zero with a. On the other hand, ft is called an infini-
tesimal of the first order with respect to a, if the ratio ft/ a
approaches a limit K different from zero as a approaches zero. In
this case
^ = /iT + e,
a
where c is another infinitesimal with respect to a. Hence
ft=a{K + t)= Ka-\- a€,
and Ka is called the principal part of ft. The complementary term
arc is ah infinitesimal of higher order with respect to a. In general,
if we can find a positive power of a, say a*, such that ft/a^
approaches a finite limit K different from zero as a approaches
zero, ft is called an infinitesimal of order n with respect to a. Then
we have
| = a: + .,
or
ft = c^{K + €)= ^o- -f a"c.
The term KoT is again called the principal part of ft.
Having given these definitions, let us consider a continuous func-
tion y =f(x), which possesses a derivative f\x). Let Ai! be an
* With the possible exception or Newton's notation. — T&AK&
20 DKRIVATIVES AND DIFFERENTIALS [I, §14
increment of x, and let Ay denote the corresponding increment of y.
From the very definition of a derivative, we have
where c approaches zero with Ax. If Ax be taken as the principal
infinitesimal, Ay is itself an infinitesimal whose principal part is
/*(x)Ax.* This principal part is called the differential of y and is
denoted by dy.
dy=f{x)^.
Whan /(x) reduces to x itself, the above formula becomes dx = Ax'y
and hence we shall write, for symmetry,
dy=f'(x)dx,
where the increment dx of the independent variable x is to be given
the same fixed value, which is otherwise arbitrary and of course
variable, for all of the several dependent
functions of x which may be under consid-
eration at the same time.
Let us take a curve C whose equation is
y=f(^)f and consider two points on it, M
and M', whose abscissae are x and x + dx,
respectively. In the triangle MTN we have
NT = MN tan Z TMN = dxf'(x).
Fio. 1 " \ /
Hence NT represents the differential dy,
while Ay is equal to NM'. It is evident from the figure that M'T
if an infinitesimal of higher order, in general, with respect to NT,
as ir approjMjhes A/, unless MT is parallel to the x axis.
Successive differentials may be defined, as were successive deriv-
ati768, each in terms of the preceding. Thus we call the differ-
ential of the differential of the first order the differential of the
second order, whore dx is given the same value in both cases, as
abore. It is denoted hy d^y.
d*y = d(dy) = \_fXx) dx-] dx =/"(x) (dx)\
SimUarly, the third differential is
rf»y = d{d^y) = [/"(a;) rfx«] dx =f"(x) (dxy,
r
>
T
a
^
^y
y
d
J
» <
} ^
i^*!!iJ!L2!r^'l7! ^u^^ **•"' "'^"'** ***' "»«« where/'(a;) = 0. It Is, how-
7^S^Ti.JTl^Ji"r?' ''^"'''"" "' ^1/ =/'(.) Ax in this case also.
•fm laoogb it U not the prinolpAl (wrt of Ay.— Traks.
I, §14] THE DIFFKKKNTIAL NOTATION 21
and so on. In general, the di£ferential of the differential of ordex
The derivatives /'(x), /"(x), • •, /^"^x), • • • can be expressed, on
the other hand, in terms of differentials, and we have a new nota-
tion for the derivatives :
To each of the rules for the calculation of a derivative corresponds
d rule for the calculation of a differential. For example, we have
c? x*" = mx^'^dXf da* = a* log a dx ;
d log X = — > d sin x = cos xdx , • • • ;
X
, . dx 7 i. ^
cf arc sin 2 = > a arc tan x =
Let us consider for a moment the case of a function of a function.
Let y =f(u), where u is a function of the independent vaiiable x.
Then ^. ,
whence, multiplying both sides by rfx, we get
yx^=/Wx ?/,'/x;
that is,
</y =f(ii)du.
The formula for rfy is therefore the same as if u were the inde-
pendent variable. This is one of the advantages of the differential
notation. In the derivative notation there are two distinct formulae,
to represent the derivative of y with respect to x, according as y is
given directly as a function of x or is given as a function of x by
means of an auxiliary function u. In the differential notation the
same formula applies in each case.*
li y = f(u, V, xv) is a composite function, we have
at least if /^,/,,/^ are continuous, or, multiplying by <ix,
y^dx = u^dxf, + v^dxf^ + tr,<&/^ ;
• This particular advantajfe Is slight, however ; for the last formula above is equally
well a general one and covers both the cases mentioned. — Trans.
vdu — udv
S2 DERIVATIVES AND DIFFERENTIALS [I, §16
ihit 18,
dy =fndu -\-f,dv -^f^dw.
Thus we have, for example,
d{uv)=udv -\-vdUf rff-j =
The same rules enable us to calculate the successive differentials.
Let us aeek to calculate the successive differentials of a function
y =/(«), for instance. We have already
dy =f\u)du.
In order to calculate <?y, it must be noted that du cannot be regarded
as fixed, since u is not the independent variable. We must then
calculate the differential of the composite function /'(w) (iw, where u
and du are the auxiliary functions. We thus find
d^y =f\u)du^ +fiu)d^u.
To calculate rf'y, we must consider «Py as a composite function, with
ti, du, d^u 2i8 auxiliary functions, which leads to the expression
d*y =f"(u) dv* + 3/"(w) dud^u +f{u) d^u ;
and so on. It should be noticed that these formulae for d^y^ d^y,
etc., are not the same as if w were the independent variable, on
account of the terms d^u, d^Uy etc.*
A similar notation is used for the partial derivatives of a function
of several variables. Thus the partial derivative of order n of
f(Xf y, «), which is represented by fxP^^r in our previous notation,
u represented by
s^Ms^' -? + ? + '• = "'
in the differential notation. t This notation is purely symbolic, and
in no sense represents a quotient, as it does in the case of functions
of a single variable.
15. Toul differentials. Let u =/(a;, y, «) be a function of the
three independent variables x, y, z. The expression
* This diMdvanUfftt would M«m completely to offset the advantage mentioned
•bore. BUiotly apeaklng, we should distinguish between dPj/ and dly, etc. — Tran«).
t This Oit of the letter d to denote the partial derivatives of a function of several
VMtehUi Is dos to JaooM. Before his time the same letter d was used as is used for
Uwi d«rivativMi of a function nf h Niti^le variable.
1,516] THE DIFFKKENTIAL NOTATION 28
is called the total differential of w, where rfx, rfy, dz are three fixed
increments, which are otherwise arbitrary, assigned to the three
independent variables x, y, z. The three products
^^' ^^^' a;^*
are called partial differentials.
The total ditferential of the second order d^utis the total differ-
ential of the total differential of the first order, the increments
dxy dy, dz remaining the same as we pass from one differential to
the next higher. Hence
or, expanding,
/ c^f av av \
rf«a>= -Idx + ^dy + -^dz)dx
\ cx^ ex cy cxcz I
-I- 2 r^ dxdy-^2 -^ dxdz-^2 ^ dy dz.
ex cy "^ ox cz Cy cz
If d^f be replaced by df^y the right-hand side of this equation
becomes the square of
We may then write, symbolically,
it being agreed that df^ is to be replaced by d^f after expansion.
In general, if we call the total differential of the total differential
of order (71 —1) the total differential of order n, and denote it by
<i"w, we may write, in the same symbolism.
\Cx Cy ^ dz J
where a/" is to be replaced by d^f after expansion ; that is, in out
ordinary notation,
ti DERIVATIVES AND DIFFERENTIALS [I, §15
where
\b the coefficient of the term a^b'fif in the development of (a + J H-c)".
For, suppose this formula holds for d" w. We will show that it then
holds for <f*"^*«; and this will prove it in general, since we have
Already proved it for n = 2. From the definition^ we find
whence, replacing ^+y by ^/"^S the right-hand side becomes
Hence, using the same symbolism, we may write
Note, Let us suppose that the expression for G?a>, obtained in any
way whatever, is
(7) dia = Pdx-h Qdy + Rdz,
where P, Q, R are any functions x, y, «. Since by definition
we must have
where dxy dy, dz are any constants. Hence
The tingle equation (7) is therefore equivalent to the three separate
aquatuma (8)i and it determineH all three partial derivatives at once.
1,516] THK DIFFERENTIAL NOTATION 25
In general, if the nth total differential be obtained in any way
whatever, . _,^ j „j ^ , .
then the coefficients C^^ are respectively equal to the corresponding
nth derivatives multiplied by certain numerical factors. Thus all
these derivatives are determined at once. We shall have occasion
to use these facts presently.
16. Successive differentials of composite funaions. Let <d = F(u, v, w)
Ix; a composite function, //, v, w being themselves functions of the
independent variables a?, y, «, t. The partial derivatives may then be
written down as follows :
dx du dx dv dx dw dx
do»
du>
dz
__dFdu 'dFdv
du dy dv dy
_dFdu dFdv
du dz dv dz
, dFdw
CW cy
dFdw
dw dz
da>
dt
dFdu dFdv
~ du dt^ dv dt
,^dw
dw dt
If these four equations be multiplied by
and added, the left-hand side becomes
dx, dy, dz,
dtf respectively,
^<i> , , dio . , dto . du) .
— dx + -^ dy + -^ dz ■{- -^dt,
dx dy ^ dz <^y
that is, d(o J
and the coefficients of
dF dF
du dv
dF
dw
on the right-hand side are du, dv, dw, respectively. Hence
dF dF dF
(9) c?a) = T— du -f -— rfr -f ^— dw,
^ ^ CU CV CIV
and ive see that the expression of the total differential of the first
order of a composite function is the same as if the auxiliary function*
ivere the independent variables. This is one of the main advantages
of the differential notation. The equation (9) does not depend, in
form, either upon the number or upon the choice of the independent
variables ; and it is equivalent to as many separate equations as
there are independent variables.
To calculate d^u), let us apply the rule just found for dut, noting
that the second membt»r of (9) involves the six auxiliary functions
u, v, w, du, dr, die. We thus find
26 DERIVATIVES AND DIFFERENTIALS [I, §16
rf«« = -__. rf|4« 4- — — - du dv + r-TT- dudw-{--r- d*u
• ^* dudv ducw du
'^'r—r-dudw-k"r-r-dvdw+ |-t c?«;^ + ^ cPw,
or, simplifying and using the same symbolism as above,
d^^^f^^du-^-^dv^^dwj ^^d^u+^d^^^d^u:.
This formula is somewhat complicated on account of the terms in
</*u, d*Vy d^Wj which drop out when u, v, w are the independent
variables. This limitation of the differential notation should be
borne in mind, and the distinction between d^ta in the two cases
carefully noted. To determine d^my we would apply the same rule
to (^<o, noting that cT'o) depends upon the nine auxiliary fimctions
tf, r, IT, du, du, dw, d^u,d^v, d^w\ and so forth. The general expres-
sions for these differentials become more and more complicated ;
</*•» is an integral function of du, dv, dw, d^u, • • •, d'^Ujd^v, d!^w, and
the terms containing d:*u, d^v, d^w are
•r- (Tw 4- -r- dPv + ^— d^w.
ou cv ow
If, in the expression for c?" w, u, v, w, du, dv, dw, • • • be replaced by
their values in terms of the independent variables, d^m becomes an
integral polynomial in dx, dy, dz, • • • whose coefficients are equal
(cf. Note, § 15) to the partial derivatives of w of order n, multiplied
by certain numerical factors. We thus obtain all these derivatives
at once.
Suppose, for example, that we wished to calculate the first aqd
second derivatives of a composite function <u=/(z*), where w is a
function of two independent variables w = <^ (x, y). If we calculate
these derivatives separately, we find for the two partial derivatives
of the first order
d(i> dutdu
dto dto du
dx du dx
dy du dy
ao)
Again, taking the derivatives of these two equations with respect
to X, and then with respect to y, we find only the three following
distinct equations, which give the second derivatives:
Kun
THE DIFFERENTIAL NOTATION
27
(11)
dx* " du*\dx)
du
d*i
d*ot du du dm
d^io _ d*ia/du
[ 'd^ '"d^^\d^
)* ,dm d^
"*■ du d/
The second of these equations is obtained by differentiating the
first of equations (10) with respect to y, or the second of them with
respect to x. In the differential notation these five relations (10)
and (11) may be written in the form
(12)
dm =
du
du,
M 3*u> , , , dm j^
du'
du
If du and d*u in these formulae be replaced by
du dll
^dx-^-dy and
d^
dx
ldx'-^2
C^u
a«i
dxdy'^'^'^y^-d^-^^^
redpectively, the coefficients of dx and dy in the first give the first
partial derivatives of w, while the coefficients of dx^, 2dxdyj and
dy^ in the second give the second partial derivatives of m.
17. Differentials of a product. The formula for the total differential
of order n of a composite function becomes considerably simpler
in certain .special cases which often arise in practical applications.
Thus, let us seek the differential of order n of the product of two
functions w = mv. For the first values of n we have
dii) = V du -{- u dr, d^w = vd^u -\- 2dudv -\- ud^v, •••;
and, in general, it is evident from the law of formation that
rf"<i) = vd^u H- Cidvd''-Ui -\- C^d^vdr-'u ^ \-u^Vj
where C,, C„ • • are positive integers. It might be shown by alge-
braic induction that these coefficients are equal to those of the
expansion of (a -f- />)* ; but the same end may be reached by the
following method, which is much more elegant, and which applies
to many similar problems. Observing that Cj, C,, • • • do not depend
upon the particular functions u and v employed, let us take the
88 DERIVATIVES AND DIFFERENTIALS [I, §17
special functions u = <?*, v = e», where ar and y are the two inde-
pendent variables, and determine the coefficients for this case. We
thus find
du-e'dxy d^u = e'dx^, ••,
dv = e^dy, d^v = e^dy^ • • • ;
and the general formula, after division by e^+^ becomes
{dx 4- dyy = dx* + C^dydx""-^ + C^dfdx''-'^ -\ -{■ d^.
Since dx and dy are arbitrary, it follows that
n ^ n(n-l) n(n -1) "-jn -p + V)
^«==r •" 1.2 ' "' ^"" 1.2...J9
and consequently the general formula may be written
(n)dr(uv) = vdru+'^dvd^-Ui + '^^^^~^ ^ d^vd^-^'u + '-' + ud^v.
This formula applies for any number of independent variables.
In particular, if u and v are functions of a single variable x, we
have, after division by rfa;*, the expression for the wth derivative of
the product of two functions of a single variable.
It is easy to prove in a similar manner formulae analogous to
(13) for a product of any number of functions.
Another special case in which the general formula reduces to a
simpler form is that in which w, v, w are integral linear functions
of the independent variables x, y^ z.
u= ax-h by+ cz+f ,
v = a'x -\- b'y + c'z +/' ,
w = a"x + b"y + c"z -[-/",
where the coefficients o, a', a", b, 6', • • • are constants. For then we
have
du= adx -{- bdy -\' cdzy
dv = a'dx -f b'dy -f- c'dz,
dw = a"dx-^b"dy + c"dz,
and all the differentials of higher order d^u, d^Vy dTw, where n> 1,
ranish. Hence the formula for <i"ca is the same as if w, v, w were
the independent variables ; that is,
I, J 18] THE DIFFERENTIAL NOTATION 29
. (dF , dF . dF , Y>
(irui = \-;r-du-\--r-dv-\--r- aw I .
\rit cv cw I
We proceed to apply this remark.
18. Homogeneous functions. A function ^(x, y, «) is said to be
homogeneous of degree m, if the equation
is identically satisfied when we set
u — tx^ V =.tiiy w = tz.
Let us equate the differentials of order 7i of the two sides of this
equation with respect to t, noting that u, v, w are linear in f, and that
du = X dty dv = y dty dw = « dt.
The remark just made shows that
y'^'^'^dv'^^^J = m(m -1) . .. (m - n -hl)r-^(x, y, z).
If we now set / = 1, w, v, w; reduce to x, y, «, and any term of
the development of the first member,
becomes
'^dx^di/^dz'-'^''^''^'
whence we may write, symbolically,
y'di'^'^d^'^''^) "^ "'('^ - 1) • • • (m - n+l)<t>{x, y, z),
which reduces, for n = 1, to the well-known formula
. , X ^<l> ^4> ^4>
Various notations. We have then, altogether, three systems of nota-
tion for the partial derivatives of a function of several variables, —
that of Leibniz, that of Lagrange, and that of Cauchy. Each of
these is somewhat inconveniently long, especially in a complicated
calculation. For this reason various shorter notations have been
devised. Among these one first used by Monge for the first and
80 DERIVATIVES AND DIFFERENTIALS [I, §19
•eoond derivatives of a function of two variables is now in common
use. If « be the function of the two variables x and y, we set
d% dz d^z _ d^z ._?!f.
^"^' ^"a^' '■"a^^' '"dxdy ^"dy^'
md the total differentials dz and cPz are given by the formulae
dz= pdx-\-qdi/,
d^z = rdx^ -\- 2 s dx dy -\- 1 dy^.
Another notation which is now coming into general use is the
following. Let « be a function of any number of independent vari-
ables Xi, Xj, x», •••,«» J then the notation
^'^"*"'"*^ dx'^dxt^'-dx'^
is used, where some of the indices ^i, a^, • • •, a„ may be zeros.
19. Applications. Let y =f{x) be the equation of a plane curve C with
reqiect to a set of rectangular axes. The equation of the tangent at a point
Mix, V) is
Y-y = y'{X-x).
The dope of the normal, which is perpendicular to the tangent at the point of
tangency, is — l/y' ; and the equation of the normal is, therefore,
{T-y)y'+{X-x)=:0.
Let P be the foot of the ordinate of the point 3f, and let T and N be the
points of intersection of the x axis with the tangent and the normal, respectively.
The distance PN is called the subnormal ;
Pr, the subtangent; MN, the normal; and
MT, the tangent.
From the equation of the normal the ab-
scissa of the point iV is x + yy\ whence the
subnormal is ± yy\ If we agree to call the
length PN the subnormal, and to attach the
sign + or the sign — according as the direc-
tion PN is positive or negative, the subnormal
will always be yy' for any position of the curve
C. Likewise the subtangent is — y/y\
The leogthe MN and MT are given by the triangles MPN and MPT:
MNz^y/Mp^PN^^yVTTy^,
MT = V5p« + pT* = ^VTT?^,
.-T* ***** pwN«ni may be given regarding these lines. Let us find, for
iMlftDet, an the eorvee for which the subnormal is constant and equal to a given
ombtr a. This amounta to finding all the functions y =/(x) which satisfy
the eqoaUoa loT = a. The left-hand side U the derivative of yV2, while the
I, Em.] EXKRCISES 81
right-hand side ia the derivative of ax. These functions can therefore diflsr
only by a constant ; whence
which is the equation of a parabola along the z axis. Ag|iin, if we seek the
curves for which the subtangent is constant, we are led to write down the eqa»-
tion ]/ /y — \/a\ whence
X *
logy = - + log C, or y = Ce^,
a
which is the e(}uaunn of a transcendental curve to which the x axis is an asymp-
tote. To find the curves for which the normal is constant, we have the equation
The first member is the derivative of — Va^ — y« ; hence
- Vaa-y« = x + C,
or
(X + C7)9 + 1/2 = a2,
which is the equation of a circle of radius a, whose center lies on the x axis.
The curves for which the tangent is constant are transcendental curves, which
we shall study later.
Let y = /(x) and Y = F{x) be the equations of two curves C and C, and let
Af, M' be the two points which correspond to the same value of x. In order that
the two subnormals should have equal lengths it is necessary and suflficient that
yr'=±yy';
that is, that Y^ = ± y^ i- C, where the double sign admits of the normals' being
directed in like or in opposite senses. This relation is satisfied by the curves
and also by the curves
.« = >-<,^. r« = ^,
a^ a^
which gives an easy construction for the normal to the ellipse and to the hyperbola.
EXERCISES
1. Let p = f{d) be the equation of a plane curve in polar coordinates. Through
the pole draw a line perpendicular to the radius
vector OM, and let T and N be the points where this
line cuts the tangent and the normal. Find expres-
sions for the disunces OT, ON, MN, and MT in
terms of /(<?) and f\e).
Find the curves for which each of these distances,
in turn, is constant.
2. Let y = /(x), t = (x) be the equations of a
skew curve r, i.e. of a general space curve. Let N Fio. 3
S2 DERIVATIVES AND DIFFERENTIALS [I, Exs.
to the point where the normal plane at a point 3f, that is, the plane perpendicu-
lar to Um tangent at JT, meet« the z axis ; and let P be the foot of the perpen-
dlcaUr from Jf to the « axis. Find the curves for which each of the distances
PN and MN, in torn, is constant.
[NoU. The« cures lie on paraboloids of revolution or on spheres.]
S. IMennine an integral polynomial /(x) of the seventh degree in z, given
that/(x) + 1 is divisible by (» - 1)* and f{x) - 1 by (x+1)*; Generalize the
problem.
4. Show that if the two integral polynomials P and Q satisfy the relation
vT^r^=QVl-x2,
dP ndx
Vl - pa Vl -x2
» is a positive integer.
[ JTote. From the relation
(a) l-P2=Q2(i_a;2)
it follows that
(b) -2PP'=Q[2Q'(l-x2)-2Qx].
The equation (a) shows that Q is prime to P ; and (b) shows that P' is divisible
5e. Let A (z) be a polynomial of the fourth degree whose roots are all dif-
ferent, and let X = IT/ F be a rational function of t, such that
^' Q{t) ^'
where Ri (0 is a polynomial of the fourth degree and P/ Q is a rational function.
Show that the function U/ V satisfies a relation of the form
dx kdt
VR{x) VRiit)
wbare l( is a constant [Jacobi.]
INoU. Each root of the equation R(U/ V) = 0, since it cannot cause R'{x)
to vanish, most cause UV - VU\ and hence also dx/db, to vanish.]
••. Show that the nth derivative of a function y = 0(m), where u is a func-
tion of the independent variable x, may be written in the form
W 1^ = ^i ♦'(«) + Y^ ♦"(") + ■• • + 7-^ «<")(»).
wbart
(b)
da^ ' ' 1.2^ ' ' 1.2-
d»tt» Ac^ d''u*-^ fc(Jk-l) jd»u*2*
da^ l" dx» 1.2 ^ dx» "^
+ (-l)»-tfcu*-i^ (fc=l,2....,n).
[Ill* DOtlM that the nth derlvatiTe may be written in the form (a), where the
oodBdenU At. A*. . A^ are independent of the form of the function <f>{u).
I, Exb] exercises 83
To find their values, set ^(u) equal to u, u*, • • •, u" successively, and solve the
resulting equations for ^i, At, • • • , A^. The result is the form (b).]
7*. Show that the nth derivative of <p{x^ is
^^^^^^ = (2x)-^(-)(x«) + n(n- l)(2x)"-»'-f"-"f -2, , . ..
+ "<" - ')•-(" -^P+ ») (2x)»-'.-».-..(x') + ■■■.
1 . 2' • 'P
where p varies from zero to the last positive integer not greater than n/2, and
whore 0<')(x'') denotes the tth derivative with re8pect to x.
Apply this result to the functions e-**, arc sin x, arc tan x.
8*. If X = cos u, show that
d«-»(l-x«)'"-* , ,, . ,1.3.6...(2m-l) ,
i = (- l)"'-! i i sin mu.
dx"«-» m
[OlIKDK RODRIUUEA.]
9. Show that Legendre's polynomial,
^ 1 d»
2 . 4 . 6 • • . 2 n dx"
satisfies the differential equation
Hence deduce the coefBcients of the polynomial.
10. Show that the four functions
yi = sin (71 arc sin x), ys = sin (n arc cos x),
ya = cos (n arc sin x), y^ = cos (n arc cos x),
satisfy the differential equation
(1 -x2)y"-xy' + n2y = 0.
Hence deduce the developments of these functions when they reduce to poly-
nomials.
11*. Prove the formula
1
d" 1 e*
_(x.-.e.) = (-l).— .
[Halphen.]
12. Every function of the form 2 = x ^ (y /x) + ^ (y /x) satisfies the equation
rx« + 2 sxy + ty^ = 0,
whatever be the functions and ^.
13. The function z = x0(x + y) + yt/'(x + y) satisfies the equation
r-2a + t = 0,
whatever be the functions </> and ^.
S4 DERIVATIVES AND DIFFERENTIALS [I, Ei&
14. The function s = /[x + 4> (y)] aatisfies the equation pa = qr, whatever
be the functions/ and ^.
15. The function « = x"^(y/x) + jr"^(y/«) satisfies the equation
rx» + 2«cy + «y2 + px + gy = n^z,
whaterer be the functions and ^.
16. Show that the function
y = \x - ai\th{x) + \x - ai\<h{x) -\- '•• + \x - an\<f>n{x),
where ^ (z), ^ (x), • • • , 0» (x), together with their derivatives, 0i (x), 03 (x), • • • ,
^ (X), are continuous functions of x, has a derivative which is discontinuous
forx = Oi, Os, •••, a..
17. Find a relation between the first and second derivatives of the function
t =/(Xi, u), where u = 0(xt, Xj); Xi, Xa, Xz being three independent variables,
and /and two arbitrary functions.
18. Let/(x) be the derivative of an arbitrary function /(x). Show that
1 d^u _ 1 d^v
udx^ ~ V dx2'
where u = [f(x)]-i and » =/(x) [/'(x)]-i.
,19«. The nth derivative of a function of a function u = (y), where y = ^ (x),
may be written in the form
1^0 =
the sign of summation extends over all the positive integral solutions of
the equation i + 2j + 3 A + • • • + iA: = n, and where p = i+j + "• + k.
(Faa d» Bbumo, Quarterly Journal of Mathematics, Vol. I, p. 359.]
CHAPTER II
IBfPLICIT FUNCTIONS FUNCTIONAL DETERMINANTS
CHANGE OF VARIABLE
L IMPLICIT FUNCTIONS
20. A particular case. We frequently have to study functions for
which no explicit expressions are known, but which are given by
means of unsolved equations. Let us consider, for instance, an
equation between the three variables ar, y, «,
(1) F{x,y,z) = 0.
This equation defines, under certain conditions which we are about
to investigate, a function of the two independent variables x and y.
We shall prove the following theorem :
Let X — Xq, y = 1/oy ^ = ^o ^^ ct set of values which satisfy the eqtm-
tion (1), and let us suppose that the function F, together with its first
derivatives, is continuous in the neighborhood of this set of values.*
If the derivative F, does not vanish for x = Xq, y = yoi z = Zq, there
exists one and only one continuous function of the independent variables
X and y which satisfies the equation (1), aiid which assumes the value Zq
when X and y assume the values Xq and y^, respectively.
The derivative F, not being zero for a; = a;©, y = yoj ^ = «o> let us
suppose, for definiteness, that it is positive. Since F, F„ F^, F, are
supposed continuous in the neighborhood, let us choose a positive
number / so small that these four functions are continuous for all
sets of values «, y, z which satisfy the relations
(2) |a;-xo|</, \y-yo\^ly \z-z,\<l,
and that, for these sets of values of aj, y, «,
^.(a^, y, «) > F,
• In a recent article {Bulletin de la Soci^U Math^matique de France, Vol. XXXI,
1903, pp. 184-192) Groursat has showTi, by a method of successive approximations, that
it is not necessary to malce any assumption whatever repardinjj F, and F^, even as to
their existence. His proof makes no use of the existence of F, and f\. His general
theorem and a sketch of his proof are given in a footnote to § 25. — Trans.
36
ge FUNCTIONAL RELATIONS [II.§20
whm P is some positive number. Let Q be another positive num-
ber greiOer than the absolute values of the other two derivatives
F-, F, in tlie same region.
Giving X, y, « values which satisfy the relations (2), we may then
write down the following identity :
F(x,y,M)^F{x,,yo,z,)^F{x,y,z)-F{x,, y, z)^F{x,, y, z)
-F(xo, yo, z) + F(xo, 2/0, z)-F{xo, yo, «o) ;
or, applying the law of the mean to each of these differences, and
obeerring that F(xo, y©, «o) =0,
F(x,y,z)= (x-Xo)F,lxo + e(x-Xo), y, «]
+ (y - yo) F^ [xo, 2/0 + e\y - yo), «]
+ (z- Zo) F, [Xo, yo, «o + e"(z - «o)].
Hence F(x, y, «) is of the form
,«v i '•'(*» y» «) = ^ (^' 2/, «) (^ - ^o)
W ^ +^(x, y, «)(y-yo) + C'(a;, y, «)(«-«o),
where the al)solute values of the functions A(Xy y, «), 5(x, y, ^),
C(x, y, «) satisfy the inequalities
\A\<Q, \B\<Q, \C\>P
for all sets of values of x, y, sj which satisfy (2). Xow let c be a
positive number less than I, and -q the smaller of the two numbers
/ and P€/2Q. Suppose that x and y in the equation (1) are given
definite values which satisfy the conditions
and that we seek the number of roots of that equation, z being
regarded as the unknown, which lie between Zq — e and Zq + «. In
ftfl the expression (3), for F(x, y, z) the sum of the first two terms is
always less than 2Qyi in absolute value, while the absolute value of
the third term is greater than Pt when z is replaced by Zq ± «. From
the manner in which tf was chosen it is evident that this last term
determines the sign of F. It follows, therefore, that F(x, y, Zq — «) <
and F(Xf y, «© -f <) > ; hence the equation (1) has at least one root
which lies between «o — « and Zq + e. Moreover this root is unique,
•ioce the derivative F, is positive for all values of z between Zq — e
and «t 4- «. It is therefore clear that the equation (1) has one and
only one root, and that this root approaches Zq as x and y approach
x^ and y«, respectively.
II, $20] IMPLICIT FUNCTIONS 87
Let us investigate for just what values of the variables x and y
the root whose existence we have just proved is defined. Let h be
the smaller of the two numbers I and 1H/2U\ the foregoing reason-
ing shows that if the values of the variables x and y satisfy the
inequalities |x — a"o| < A, |y — yd < ^) ^^^ equation (1) will have one
and only one root which lies between z^ — I and Zq -h /. Let 72 be a
square of side 2 A, about the point Moix^^ y,,), with its sides parallel
to the axes. As long as the point (x, y) lies inside this square,
the equation (1) uniquely determines a function of x and y, which
remains between Xq^I and Zq-\- I. This function is continuoos, by
the above, at the point Mq^ and this is likewise true for any other
point Ml of R ; for, by the hypotheses made regarding the func-
tion F and its derivatives, the derivative /•\(xi, yi, z^) will be posi-
tive at the point M^y since \x-^—Xq\<1, \yi— yn\<lj |«i— «o|<^-
The condition of things at ^fl is then exactly the same as at A/^,
and hence the root under consideration will be continuous for
x^Xi, y = yi.
Since the root considered is defined only in the interior of the
region /?, we have thus far only an element of an implicit function.
In order to define this function out-
side of R, we proceed by successive
steps, as follows. Let /. be a con-
tinuous path starting at the point
(^0) //o) and ending at a point (A', }')
outside of R. Let us suppose that
the variables x and y vary simul-
taneously in such a way that the —^
point (x, y) describes the path L. pj^^ ^
If we start at (xq, y^ with the value
Zq of z, we have a definite value of this root as long as we remain
inside the region R. Let M^ {x^, y{) be a point of the path inside i?,
and zi the corresponding value of z. The conditions of the theorem
being satisfied for x =Xu y = yi, z = z^, there exists another region
Riy about the point M^ inside which the root which reduces to «, for
X = Xi, y = yi is uniquely determined. This new region /?, will
have, in general, points outside of R. Taking then such a point Aft
on the path L, inside R^ but outside R, we may repeat the same con-
struction and determine a new region R^, inside of which the solu-
tion of the equation (1) is defined; and this process could be
repeated indefinitely, as long as we did not find a set of values of
X, y, z for whigh F, = 0. We shall content ourselves for the present
y)
88 FUNCTIONAL RELATIONS [n,§21
with these statements ; we shall find occasion in later chapters to
tieat oertain analogous problems in detail.
8L DeriTttiTes of implicit functions. Let us return to the region
Hf and to the solution z = <t>(x, y) of the equation (1), which is a
continuous function of the two variables x and y in this region.
This function possesses derivatives of the first order. For, keeping
y fixed, let us give x an increment Aa;. Then z will have an incre-
ment Lx, and we find, by the formula derived in § 20,
F{x + Aar, y, « + A«) - F{x, y, z)
e AxF,(a: + dAx, y, « -f A«) + A«F,(aj, y, « + ^'A«) = 0.
Henoe
Ag_ j;(a;+gAa;, y,z + £^z) ,
Aa? F^{x, y, z + $' Lz)
and when Ax approaches zero, ^z does also^ since ;;; is a continuous
function of x. The right-hand side therefore approaches a limit,
and z has a derivative with respect to xi
dx F,
In a similar manner we find
dz__ F^
dy- fJ
Note. If the equation F = is of degree m in «, it defines m
functions of the variables x and y, and the partial derivatives dz/dxy
dz/dy also have m values for each set of values of the variables
X and y. The preceding formulae give these derivatives without
ambiguity, if the variable z in the second member be replaced by
the value of that function whose derivative is sought.
For example, the equation
«* -f- y* + «* - 1 =
the two continuous functions
+ VI - aj« - y* and - Vl - «« - y»
for Talues of x and y which satisfy the inequality x* -^ y^ < 1.
The first partial derivatives of the first are
— » — y
Vl - ar« - y«* Vl - x« - y«'
n,f22] IMPLICIT FUNCTIONS 39
and the partial derivatives of the second are found by merely chang-
ing the signs. The same results would be obtained by using the
formula
dz ^ X cz _ y
dx z dy z
replacing t by its two values, successively.
22. Applications to surfaces. If we interpret x, y,zaA the Cartesian
coordinates of a point in space, any equation of the form
(4) F(x,y,«) =
represents a surface S. Let (xq, y^, s;^) be the coordinates of a point
A of this surface. If the function F, together with its first deriva-
tives, is continuous in the neighborhood of the set of values Xq, y^f «„»
and if all three of these derivatives do not vanish simultaneously
at the point .1, the surface .s' has a tangent plane at A. Suppose,
for instance, that F, is not zero for ar = Xq, y = y^j « = «©• Accord-
ing to the general theorem we may think of the equation solved
for z near the point A , and we may write the equation of the surface
in the form
z = <i>(x, y\
where <f> (x, y) is a continuous function ; and the equation of the
tangent plane at A is
Replacing dz/dx and dz /dy by the values found above, the equation
of the tangent plane becomes
<« (S).<--'-(l).<'-->-(tl<-->=»-
If F, = 0, but Fj ^ 0, at A^ we would consider y and z as inde-
pendent variables and x as a function of them. We would then
find the same equation (5) for the tangent plane, which is also evi-
dent a priori from the symmetry of the left-hand side. Likewise
the tangent to a plane curve F(x, y) = 0, at a point (x^, y^), is
<-- •)(£)/<-'->®).-»'
If the three first derivatives vanish simultaneously at the point A,
40 FUNCTIONAL RELATIONS [11,523
the preceding reasoning is no longer applicable. We shall see later
(Chapter III) that the tangents to the various curves which lie on
the surface and which pass through A form, in general, a cone and
not a plane.
In the demonstration of the general theorem on implicit functions
we assumed that the derivative F^ did not vanish. Our geometrical
intuition explains the necessity of this condition in general. For,
if jc = but F^ ^ 0, the tangent plane is parallel to the z axis,
and a line parallel to the z axis and near the line x = x^^, y =^ y^
meets the surface, in general, in two points near the point of
tangency. Hence, in general, the equation (4) would have two
roots which both approach Zq when x and y approach x^^ and 2/g,
respectively.
If the sphere x* + y* + «' — 1 = 0, f or instance, be cut by the line
y = 0, X = 1 -f c, we find two values of «, which both approach zero
with < ; they are real if c is negative, and imaginary if c is positive.
23. Successive derivatives. In the formulae for the first derivatives,
dx" fJ dy fJ
we may consider the second members as composite functions, z being
an auxiliary function. We might then calculate the successive deriv-
atives, one after another, by the rules for composite functions. The
existence of these partial derivatives depends, of course, upon the
existence of the successive partial derivatives of F(x, y, z).
The following proposition leads to a simpler method of determin-
ing these derivatives.
If several functiona of an independent variable satisfy a relation
F = 0, their derivatives satisfy the equation obtained by equating to
tiero the derivative of the left-hand side formed by the rule for differ-
mHaiimg composite functions. For it is clear that if F vanishes
identioally when the variables which occur are replaced by func-
tions of the independent variable, then the derivative will also van-
ish identically. The same theorem holds even when the functions
which satisfy the relation /•' = depend upon several independent
variables.
Now suppose that we wished to calculate the successive derivatives
of an implicit function y of a single independent variable x defined
by the relation
U.pj IMPLICIT FUNCTIONS 41
We find successively
d^F d*F d*F dF
S'F , . e^F , . „ S*F „ , ^ 8*F „ e*F .
+ ^I^y'^" + ¥y^"' = '''
from which we could calculate successively y', y", y'", • ••.
Example. Given a function y =/(x), we may, Inversely, consider y as the
independent variable and z as an implicit function of j/ defined by the equation
y=/{x). If tlie derivative /'(x) does not vanish for the value asot where
yo = /(Zo), there exists, by the general theorem proved above, one and only one
function of y which satisfies the relation y =/(x) and which takes on the value
Xo for y = yo- This function is called the inverse of the function /(z). To cal-
culate the successive derivatives z„, z,^, z^, • • • of this function, we need merely
differentiate, regarding y as the independent variable, and we get
1 = /'(Z) Zy,
= /"(z) (z,)'» + /'(x)z^,
= r\x) (z,)« + 3/"(z) x,x^ + /'(z) z/,
whence
It should be noticed that these formulae are not altered if we exchange z, and
/'(z), Xj/i and /"(z), Zy« and /'"(z), • • • , for it is evident that the relation between
the two functions y = /(z) and z = ^ (y) is a reciprocal one.
As an application of these formulae, let us determine all thoee function*
y=.f(x) which satisfy the equation
y'y"' - 3y"'' = 0.
Taking y as the independent variable and z as the function, Uiis equation
becomes
But the only functions whose third derivatives are zero are polynomials of at
most the second degree. Hence z must be of the form
z= Ciy«+ Cty-\- C,
where Ci, Cj, Cg are three arbitrary constants. Solving this equation for y,
we see that the only functions y = /(z) which satisfy the given equation an
of the form ^____„
y = o i Vto + c,
42 FUNCTIONAL RELATIONS [II, §24
when a, ft, c are three arbitrary constants. This equation represents a parabola
wboM axis is parallel to the z axis.
M. PutUl derivatiyes. Let us now consider an implicit function
of two variables, defined by the equation
(6) JF'(x,y,«) = 0.
The partial derivatives of the first order are given, as we have seen,
by the equations
To determine the partial derivatives of the second order we need
only differentiate the two equations (7) again with respect to x and
with respect to y. This gives, however, only three new equations,
for the derivative of the first of the equations (7) with respect to y
is identical with the derivative of the second with respect to x.
The new equations are the following:
d^F d^F dz d^F /dzV dF d'^z _
/ dx^^ dx dz dy dz^ \dx) "^ dz dx". ~ "'
a«F^. a«F dz . d'^FVz . d^Fdzd^z . dF d^z
(8)
dx dy dx dz dy dy dzjdx dz^ dx dy dz dx dy '
df '^ dydzdy^ dz^ \dy) ^ dz df ~ "'
The third and higher derivatives may be found in a similar manner.
By the use of total differentials we can find all the partial deriva-
tives of a given order at the same time. This depends upon the
following theorem :
If several functwns Uy Vj w, -" of any number of independent vari-
ables x,y,x,"- satisfy a relation F = 0, the total differentials satisfy
the relation dF = 0, which is obtained by forming the total differential
of Fas if all the variables which occur in F were independent variables.
In order to prove this let F(u, v, t^;) = be the given relation between
the three functions m, r, w of the independent variables x, y, ^, t. The
first partial derivatives of m, v, w satisfy the four equations
dudr. dv dx'^ dw dx"^ *
«i« ^y ■*■ av ay ■*" dw a^ "" "'
maij IMPLICIT FUNCTIONS 43
du ex dv dx dw dx '
du dt dv dt dw dt
Multiplying these equations by dxy dy, dXf dt, respectively, and
adding, we tind
dF dF dF
^du-\--fdv\--fdw = dF=0.
du ou ow
This shows again the advantage of the differential notation, for the
preceding equation is independent of the choice and of the number
of independent variables. To find a relation between the second
total differentials, we need merely apply the general theorem to the
equation rfF=0, considered as an equation between w, v, w^ du,
dv, dw, and so forth. The differentials of higher order than the
first of those variables which are chosen for independent variables
must, of course, be replaced by zeros.
Let us apply this theorem to calculate the successive total differ-
entials of the implicit function defined by the equation (6), where
X and y are regarded as the independent variables. We find
cF dF dF
(dF , , dF , , dF ,Y^ , dF ^ .
and the first two of these equations may be used instead of the five
equations (7) and (8) ; from the expression for dz we may find the
two first derivatives, from that for d^z the three of the second order,
etc. Consider for example, the equation
which gives, after two differentiations,
Axdx -{-A'ydt/ + A"zdx = 0,
Adx*-^ A'di/^-{- A"dz^-h A"zd'z = 0,
whence
Axdx-^A'ydy,
<** = Ir.
and, introducing this value of dx in the second equation, we find
^ A {A x^ -\- A "z') dx' -h2AA 'xydx dy 4- A '(A 'y*-\-A "z') d/
41 FUNCTIONAL RELATIONS [11, §24
Using Mongers notation, we have then
Ax A^y
A(A7^±A^ _ AA'xy A 'jA'y' -h A"z^)
''■ A"^z* *~ A"^z*' A"^z^
This method is evidently general, whatever be the number of the
independent variables or the order of the partial derivatives which
it is desired to calculate.
JBEampte. Let t =/(x, y) be a function of x and y. Let us try to calculate
Um diflerentiaLB of the first and second orders dx and d^x, regarding y and z as
the independent variables, and x as an implicit function of tiiem. First of all,
we have
dz = ^-idx+?^dy.
dz dy
y and z are now the independent variables, we must set
d^y = d^z = 0,
and consequently a second differentiation gives
= ^dx^ + 2f^dxdy + ^dy^ + ^d:ix.
ax* dxdy dy^ dx
In Mongers notation, using p, g, r, s, t for the derivatives of f{x, y), these
•qoaUonB may be written in the form
dz = pdx + qdy^
= rdx^ + 28dxdy + tdy^ +pd^z.
Pfom the first we find
^^ dz-qdy ^
P
and, fttbttituting this value of dx in the second equation,
^j^^__rd^-^2{pa^qr)dydz-\-(qir-2pq8-{-pH)dy^
The flrat and second partial derivatives of x, regarded as a function of y and
S, tharalbra, have the following values :
5? -1 ax _ q
dz ""p' dy~ p'
— « - I., -^ = qr-p a dix _ 2pq8-pH-q^r
«■• p«' dydx p^ ' dy^~ pi
As an appllcaUon of theM formula, let us find all those functions /(x. y\
which aaUafy the equation ^ v » i'/
U, In the «|Qaaon i =/(x, y), x be considered as a function of the two inde-
varUblMi y and i, the given equation reduces to Xy> = 0. This means
n.j2a]
IMl'LICIT FUNCTIONS
45
that Xy \b independent of y ; and hence x, = 0(2)« where 0(c) Is an arbitrary
function of z. Thia, in turn, may Ik« \vri»t.... in the form
^
[x - y (p[Z)j - 0,
which shows that z — y 0(2) is Independent of y. Hence we may write
« = y0U) + ^(«),
where \J/ (z) is another arbitrary function of 2. It is clear, therefore, that all the
functions z =/(x, y) which satisfy the given equation, except those for which /»
vanishes, are found by Kolving this last equation for 2. This equation represents
a surface generated by a straight line whicli is always parallel to the xy plane.
25. The general theorem. Z^et us consider a system ^f n equations
^i(«i, arj, •••, a-p; t/,, Wj, •■•, wj = 0,
(E)
, j-p , 1*1, i*„
Xpy Wl, W,,
P*
Suppose
between the n -\- p variables Wi, i^j, ••, w„; ari, x„ • ••, ar^
^Aa^ ^Aese equations are satisfied for the values a*j = a^,
Wj = mJ, •• •, u^ = ?^J; ^Aa^ the functions F, are continuous and possess
first partial derivatives which are continuous, in the neighborhood of
this system of values; and, finally y that the determinant
dui dui du^
A = dui du^ cu^
does not vanish for
dui du^
du.
Xi = X?,
u, = < {i = 1, 2, ...,/>;/: = 1, 2, ..., n).
Under these conditions there exists one and only one system of CAm-
tinunus functions u^ = <^i(a-i, a*,, • • •, x^), • • , w^ = 4>^(xi, a-,, • •-, x^)
which satisfy the equations (E) and which reduce to «?, w5, •••, t/^,
/o'-ar, = «•», '•',Xp = xl*
•In his paper quoted above (ftn., p. 35) Goursat proves that the same conclosion
may be rraohe<l without makin>j any hypotheses whatever repanling the derivatives
cFi/txj of th«> functions F, with n>pird to the r's. Otherwise the hyiwtheses remain
exactly as state<l above. It is to be notioenl that the later thei»rems r^Farding the
existence of the derivatives of tlie (unctions 4> would not follow, however, witboat
some assumptions regarding dFi/dxj. The. proof given is ba8e<l on the followiiig
46 FUNCTIONAL RELATIONS [II, §26
The determinant A is called the Jacobian^* or the Functional Deter-
minant, of the n functions Fi, F„ • •, F, with respect to the n yari-
ables Ui,Ut," •, u,. It is represented by the notation
We will begin by proving the theorem in the special case of a
system of two equations in three independent variables a;, y, z and
two unknowns u and v.
(9) Fi(x, y, «, w, v) = 0,
(10) Fa(x,y,z,u,v) = 0.
These equations are satisfied, by hypothesis, for a; = a;©? y = yo> « = ^o>
u = Oq, r = r^, ; and the determinant
gFi aF, dFi gFa
^M ^v a^ du
does not vanish for this set of values. It follows that at least one
of the derivatives dFi/dv, dF^/dv does not vanish for these same
values. Suppose, for definiteness, that dFi/dv does not vanish.
According to the theorem proved above for a single equation, the
relation (9) defines a function v of the variables x, y, «, u,
which reduces to Vq for x = Xq, y = y^, z = «q, w = i<^. Replacing v
in the equation (10) by this function, we obtain an equation between
z, y, Xf and m,
♦(x, y, «, w) = Fa[a;, y, z, w, /(a;, y, z, w)] = 0,
toUM: Let /iizuZt,",Xp;uuUt,",u^, •••./«(«!, ««,• -.Kp; "i, t*2. ••. O ben
ffmetUnu qfthsn-i-p variables a;, and u^, which, together with the n'^ partial deriva-
tktm^i/du^,are continuous near Xi = 0, x^ =0, • •, Xp = 0, u^ =0, •., m„ = 0. If
^n functions fi and the n* derivcUives cfi/dui, all vanish for this system of values^
tkmi tk§ n equations
admii one and only one system of soliUions of the form
«l«^l(«lt«t. •••.»^), ttt = *«(«!. «i,-,a:p), •., Mn = 0,(a;i,a;a, •••,ajp),
••*•* ^» ^. •••• ♦• <w« oonh'nuotM/uncfton^ o/ fAc p variables Xi, «2, •• •, «p wAicA
•II ^pprooM lero a« (A« variables all approach zero. The lemma is proved by means of
••olUol functUmit^-) ==/,(x,,x,, ...,x^; wi--^ 4«-^ ..., ui'«-»>) (t = l, 2, ...
whan ^tsO. It to •bown that the suite of functions mS'"> thus defined approaches a
ItoilUaff fttaetkm Ui, which I) aatlsfles the given equations, and 2) constitutes the only
•olatSoo. Tb* pMMfe from the lemma to the theorem consists in an easy transforma-
tiott at tbo •qoAtlooa (E) loto a form similar to that of the lemma. —Trans.
. * Jaoom, Ostft'f Journal, Vol. XXII.
il,§25] IMl'LICIT FUNCTIONS 47
which is satisfied for a; = ajo, y = y^* « = «o> u = tt^. Now
and from equation (9),
du du dv du
du dv du '
whence, replacing df/du by this value in the expression for d^/dtu,
we obtain
d^ D(u, v)
'du "^ dFi
dv
It is evident that this derivative does not vanish for the values ar^,,
yo> *oj ^^0* Hen ce^ the equation * = is satisfied when u is replaced
by a certain continuous function ti = <f> (x, y, «), which is equal to
i/q when x = Xq, y = y^y z = Zq\ and, replacing m by <^ (x, y, 2) in
/(x, y, «, w), we obtain for v also a certain continuous function.
The proposition is then proved for a system of two equations.
We can show, as in § 21, that these functions possess partial
derivatives of the first order. Keeping y and z constant, let us
give X an increment Ax, and let An and Av be the corresponding
increments of the functions u and v. The equations (9) and (10)
then give us the equations
"(S-)-"(S-')-(^- -■■)-»
where c, «', «", rj, rj', rj" approach zero with Ax, Aj/, Ar. It follows
that
Am
Ax
When Ax approaches zero, Am and ^v also approach zero ; and henoe
c, c', e", rj, rj', rj" do 80 at the same time. The ratio Am /Ax therefore
approaches a limit; that is, u possesses a derivative with respect to x :
48 FUNCTIONAL RELATIONS [n,§25
dF\dF^_df\dF\
du dxdvdvdx
du dv dv du
It follows in like manner that the ratio Av/Aa; approaches a finite
limit dv/dx, which is given by an analogous formula. Practically,
thaw derivatives may be calculated by means of the two equations
dx du dx dv dx '
dx du dx dv dx '
and the partial derivatives with respect to y and z may be found in
a similar manner.
In order to prove the general theorem it will be sufficient to show
that if the proposition holds for a system of (n — 1) equations, it
will hold also for a system of n equations. Since, by hypothesis,
the functional determinant A does not vanish for the initial values
of the variables, at least one of the first minors corresponding to the
elements of the last row is different from zero for these same values.
Suppose, for definiteness, that it is the minor which corresponds to
dP^/dii^ which is not zero. This minor is precisely
and, since the theorem is assumed to hold for a system of (n — 1)
equations, it is clear that we may obtain solutions of the first (n — 1)
of the equations (£) in the form
Ut = ^i(xi, x,, ...,0;^; w,), ..., w,.i = <^„_x(aJi, Xa, ••, Xp; u^),
where the functions <^, are continuous. Then, replacing Ui, •-•, i^„_,
by the functions </»,, • • -, <^^_, in the last of equations (E), we obtain
a new equation for the determination of w.,
It only remains for us to show that the derivative d^/du^ does
not vanish for the given set of values a-«, xj, • • -, xj, wj ; for, if so, we
can toWe this laet equation in the form
where ^ is continuous. Then, substituting this value of u^ in
4it ••, ^,.,, we would obtain certain continuous functions foi
11,525] IMi'LiLil FUNCTIONS 49
^11 ^1 '"> ^n-i ^^* ^^ order to show that the derivative in ques-
tion does not vanish, let us consider the equation
The derivatives ^<^,/^m„, 3<^,/^m,, ••, ^<^,_,/3m, are given by the
(n — 1) equations
(12)
dui du^ ^u^-i ^w, ^w« '
and we may consider the equations (11) and (12) as n linear equa-
tions for d4>i/du^j • ••, b<t>^_i/du^f d^/du^j from which we find
d^ D(F,, F„ ■■•, F,.,) ^ Z)(F„ F>, .-, FJ
du^ D{ic^, Ws, ..., u^.{) D(ui, Ui, •••, w,)
It follows that the derivative d^ /du^ does not vanish for the initial
values, and hence the general theorem is proved.
The successive derivatives of implicit functions defined by several
equations may be calculated in a manner analogous to that used in
the ease of a single equation. When there are several independent
variables it is advantageous to form the total differentials, from
which the partial derivatives of the same order may be found.
Consider the case of two functions u and v of the three variables
Xy y, z defined by the two equations
F(x, y, «, M, v) = Oy
*(ar, y, z, ?/, r) = 0.
The total differentials of the first order du and dv are given by the
two equations
ox oy cz cii cv
Likewise, the second total differentials d*u and d*v are given by the
equations
60 FUNCTIONAL RELATIONS [n,§26
and so forth. In the equations which give d'*u and d^v the deter-
minant of the coefficients of those differentials is equal for all values
of n to the Jacobian Z>(F, *)/D(?^, v), which, by hypothesis, does not
▼anish.
t6. iBTtnioii. Let ui, tit, • • •, u^ be n functions of the n independent vari-
*if ««» • • -I ^1 such that the Jacobian D(ui, 1*2, •• » w„)/D(xi, x^, •••,«„)
not Tuiish identically. The n equations
(IS) \ "* ~ ^^**' **'*■' ^^' ^* ~ 02(«1, «2, • • • » «n), • • • ,
inversely, asi, Xj, • • • , «„ as functions of wi, U2, • • • , Wn- For, taking any
of values x?, xj, • • •, xj, for which the Jacobian does not vanish, and
denoting the corresponding values of Ui, v^, • • •, t<M by u^, u^, • • ., wj, there
exists, according to the general theorem, a system of functions
«t = fl(Ul, U,,.-.,li„), Xi = rp2{UuU2,-",Ur,), .-•, X„ = ^„(Ui, Wj, • • • , Mn),
which satisfy (18), and which take on the values xj, xj, • • • , xj, respectively,
when ui = Uj, • • • , tt» = mJ. These functions are called the inverses of the func-
tions 01, 0s, • ••, 011, and the process of actually determining them is called
an inversion.
In order to compute the derivatives of these inverse functions we need merely
apply the general rule. Thus, in the case of two functions
u=/(x, y), « = 0(x, y),
if we consider u and v as the independent variables and x and y as inverse
funcUona, we have the two equations
du
•'£"*
'J-dy
dv =
50 , , 50 ,
dz.
dfdiP
.'Id.
c/di>'
dy =
dx dx
dfd<f>
dfdi>
dx dy
dy dx
dx dy
dy dz
Bnal
ly, the formulas
d0
a/
9z
dy
dz _
dv ~
ay
9u
a/«0
bzdy
a/a0
' dy dz
djdj
dz dy
dy dz
II, §27] IMPLICIT FUNCTIONS 61
-£* a/
dx ^ dy dx dx dy ey dx
27. Tangents to skew curves. Let us consider a curve C repre-
senteil bv tlie two e(juatioiis
(14) (/'x(a.,y,^)=U,
lFt(x, y, «)=0;
and let x^^ y^, Zq be the coordinates of a point A/© of this curve, such
that at least one of the three Jacobians
dF\dF\ _dF\dJ\ dF\ dF\ _ dF\ dF\ dF^ gF, cF^ dF^
dy dz dz dy dz dx dx dz dx dy dy dx
does not vanish when x, y, z are replaced by a-Q, yo> ^o> respectively.
Suppose, for definiteness, tliat />>(Fi, F^/ D{y, z) is one which does
not vanish at the point 3/o. Then the equations (14) may be solved
in the form
y = 4>(x), z = ^(x),
where <f> and ^ are continuous functions of x which reduce to y^ and
«o, respectively, when x = Xq. The tangent to the curve C at the
point Mq is therefore represented by the two equations
X-x, ^ Y-y, ^ Z-z^
1 ^\x,) f(xo)'
where the derivatives <^'(a-) and •/''(•^) ^^y ^® found from the two
equations
g+if *'<'>-!? ♦■<-)-«.
S^ + |^*V).gf(.)...
In these two equations let us set a? = x,,, y = y^, « = «o, and replace
<A'(Xo) and f(Xo) by ( r - yo) / ( A' - x^) and (Z - ;ro)/(A - Xo),
respectively. The equations of the tangent then become
(16)
53 FUNCTIONAL RELATIONS [II, §28
L ^(y> *) Jo L ^(«. ^) Jo L ^(*>y) Jo
The geometrical interpretation of this result is very easy. The
two equations (14) represent, respectively, two surfaces S^ and ^a, of
which C is the line of intersection. The equations (15) represent
the two tangent planes to these two surfaces at the poirt Mq ; and
the tangent to C is the intersection of these two planes.
The formulae become illusory when the three Jacobians above all
TanUh at the point M^. In this case the two equations (15) reduce
to a single equation, and the surfaces 5i and ^2 are tangent at the
point 3/o. The intersection of the two surfaces will then consist, in
general, as we shall see, of several distinct branches through the
point J/«.
II. FUNCTIONAL DETERMINANTS
28. Fundamental property. We have just seen what an important
r61e functional determinants play in the theory of implicit functions.
All the above demonstrations expressly presuppose that a certain
Jacobian does not vanish for the assumed set of initial values.
Omitting the case in which the Jacobian vanishes only for certain
particular values of the variables, we shall proceed to examine the
very important case in which the Jacobian vanishes identically.
The following theorem is fundamental.
Ta^ Ui, f/,, • • • , u^ be n functions of the n independent variables
jt|, X,, •••, X,. In order that there exist between these n functions
a relation H (ui, w,, • • •, w,) = 0, which does not involve explicitly/ any
of the variaJjles Xi, x^t " - f x^, it is necessary and sufficient that the
fwMtional determinant,
D(ur,Ui,'",u„)
D{XijX^y-,X^)
should vanish identically.
In thj first place this condition is necessary. For, if such a rela-
tion n(it„ w„ . . ., mJ = exists between the n functions u^, u^, - • ., w^,
the following n equations, deduced by differentiating with respect to
each of the x*8 in order, must hold :
i,ij.. FUxVCTlONAL DETKKMINANTS 58
dui dz\ du^ dxi cn,^ rr^ *
dui dx^ du^ dx^ du^ dx^ "" '
and, since we cannot have, at the same time,
— = — = =?II-o
since the relation considered would in that case reduce to a trivial
identity, it is clear that the determinant of the coefficients, which is
precisely the Jacobian of the theorem, must vanish.*
The condition is also sufficient. To prove this, we shall make
use of certain facts which follow immediately from the general
theorems.
1) Let u, V, w be three functions of the three independent variables
Xj y, z, such that the functional determinant D(u, v, w)/D(x, y, z)
is not zero. Then no relation of the form
Xdu -{• fxdv + vdw =
can exist between the total differentials du, du, dwy except for
X = /x = V = 0. For, equating the coefficients of dx, rfy, dz in the
foregoing equation to zero, there result three equations for X, /x, v
which have no other solutions than X = /i = v = 0.
2) Let CO, w, V, IV be four functions of the three independent
variables ar, y, «, such that the determinant />(»«, r, w)/ D(x, y, z)
is not zero. We can then express x, i/, z inversely as functions of
u, V, w'^ and substituting these values for ar, y, z in w, we obtain
a function
O) = * (w, V, w)
of the three variables w, v, w. If by any process whatever we can
obtain a relation of the fonn
(16) dia = Pdu+ Qdv + Rdw
*A8 Professor Osgood has pointed out, the reasoning here suppoMS that the
partial dcrivative.s dll/tu^ , c*II /tuf^, • • • , ?n /^u« do not all vanish simultaneously
for any system of values which cause 11 (ti|, u^, • • -, u„) to vanish. This supposition
\h certainly justified when the relation n = is solved for one of the variables u,.
54 FUNCTIONAL RELATIONS [li,§28
between the total differentials d<o, du, dv, dw, taken with respect to the
mdependetU variables x, y, «, then the coefficients P, Q, R are equaly
reepeetiwelyt to the three first partial derivatives of^{u, v, w) :
For, by the rule for the total difFerential of a composite function
(5 16), we have
dut = T- du -{- -^ dv -\- -^ dw'j
du ov ow
and there cannot exist ady other relation of the form (16) between
du, du, du, dw, for that would lead to a relation of the form
\ du -{- fi dv + V dw = 0,
where X, /i, v do not all vanish. We have just seen that this is
impossible.
It is clear that these remarks apply to the general case of any
number of independent variables.
Let us then consider, for definiteness, a system of four functions
of four independent variables
X = Fi(x, y, z, t),
Y = F,(x, y, z, t),
Z z= F^{x, y, z, t),
r = F,{x, 7j, z, t),
(17)
where the Jacobian D(Fi, F,, F^, F^)/D(x, y, z, t) is identically
zero by hypothesis ; and let us suppose, first, that one of the first
minors, say /)(Fi, F,, F^)/D(x, y, z), is not zero. We may then
think of the first three of equations (17) as solved for x, y, z as
functioDB of X, Y,Z,t; and, substituting these values for x, y, z in
the last of equations (17), we obtain T as a function of X, Y,Z,t:
W T^4f(X,Y,Z,t).
We proceed to show that this function * does not contain the vari-
able t, that is, that d^/dt vanishes identically. For this purpose
let ui consider the determiuaut
II.§*-«J
FUNCTIONAL DETERB4INANTS
55
A =
dFy' dF\ dF\
d^ dy dx
gF, dp; dFt
dx dy dz
dF\ dF\ e^
dx dy dz
dj\ dj\ d¥\
dx dy dz
dX
dY
dZ
dT
If, in this determinant, dXy dVj dZ, dT be replaced by their values
and if the determinant be developed in terms of rfx, rfy, dz, dty it turas
out that the coefficients of these four differentials are each zero ; the
first three being determinants with two identical columns, while the
last is precisely the functional determinant. Hence A = 0. But if
we develop this determinant with respect to the elements of the last
column, the coefficient of dT'is not zero, and we obtaia a relation of
the form
dT=PdX-ir QdY-\- RdZ.
By the remark made above, the coefficient of dt in the right-hand
side is equal to d<if/dt. But this right-hand side does not contain
dtf hence d<P/dt = 0. It follows that the relation (18) is of the form
r = <i»(x, r,z),
which proves the theorem stated.
It can be shown that there exists no other relation, distinct from
that just found, between the four functions A', 1', Z, 7', independent
of Xj y, z, t For, if one existed, and if we replaced T by *(J^, K, Z)
in it, we would obtain a relation between A', Y, Z of the form
n(A', K, Z)=0, which is a contradiction of the hypothesis that
D{X^ F, Z) / D(x^ y, z) does not vanish.
Let us now pass to the case in which all the first minors of the
Jacobian vanish identically, but where at least one of the second
minors, say Z)(F,, Ft)/D(x, y), is not zero. Then the first two of
equations (17) may be solved for x and y as functions of X. y, z, t,
and tlie last two become
Z = *, (A, }', z, 0, T -. *, (A, r, z, t).
56 FUNCTIONAL RELATIONS [il.§28
On the other hand we can show, as before, that the determinant
^-II p dX
dx dy
^Jl ^Jl dY
dx dy
!^ p dz
dx oy
vanishes identically ; and, developing it with respect to the elements
of the last column, we find a relation of the form
dZ^PdX+QdY,
whence it follows that
»-'
dt
dt "'
Id like manner it can be shown that
and there exist in this case two distinct relations between the foui
functions X, Y, Z, T, of the form
Z = *i(X, F), T=^,(X,Y).
There exists, however, no third relation dittinct from these two;
for, if there were, we could find a relation between X and F, which
would be in contradiction with the hypothesis that 2)(X, Y)/D(xy y)
if not zero.
Finally, if all the second minors of the Jacobian are zeros, but
not all four functions Xy Z, F, T are constants, three of them are
functions of the fourth. The above reasoning is evidently general.
If the Jacobian of the n functions Fi, F^, •• •, F„ of the n independ-
ent variables Xi, Xj, • ••, a;,, together with all its (w — r -f- 1)- rowed
minors, vanishes identically, but at least one of the (n — r)- rowed
minors is not zero, there exist precisely r distinct relations between
the n functions ; and certain r of them can be expressed in terms
of the remaining (n — r), between which there exists no relation.
The proof of the following proposition, which is similar to the
above demonstration, will be left to the reader. The necessary and
tMfficitnt condition that n functions of n -\- p independent variables be
conmeeted by a rf lot ion which does not involve these variables is that
one of the Jaeobians of these n functions^ with respect ^o any n
II, J 28] FUNCTIONAL DETERMINANTS 57
of the independent variableSy should vanish identically. In par-
ticular, the necessary and sufficient condition that two functions
/'\(ar,, Xj, ••, X,) and /'\(xi, x,,--, x,) should be functions of each
other is that the corresponding partial derivatives dF^/dxi ^^^
dFt/dXi should be proportional.
Note. The functions F,, Fj, • • •, F. in the foregoing theorems may
involve certain other variables yi, y,, •••, y^, besides Xj, x,, ••-, x,.
If the Jacobian 7^(Fi, F,, ••, F^)//>(xi, x,, •••, x^) is zero, the
functions F,, Fj, •••, F„ are connected by one or more relations
which do not involve explicitly the variables Xj, x,, •••, x„, but
which may involve the other variables yi,.ys, •••, y,„.
Applicationa. The preceding theorem is of great importance. The funda-
mental property of the logarithm, for instance, can be demonstrated by means
of it, without using the arithmetic definition of the logarithm. For it is proved
at the beginning of the Integral Calculus that there exists a function which Is
defined for all positive values of the variable, which is zero when x = 1, and
whose derivative is \/x. Let/(x) be this function, and let
"=/(Jj)+/(y). o = xy.
Then
X y
V £
0.
Henc& there exists a relation of the foriu
/(aj)+/(y) = 0(«y);
and to determine <t> we need only set y = 1, which gives f{x) = ^ (z). Hence,
since x is arbitrary,
f{x)-^f{y)=f{xy).
It is clear that the preceding definition might have led to the discovery of
the fundamental properties of the logarithm had they not been known before the
Integral Calculus.
As another application let us consider a system of n t-ciuations in n unknowns
(19)
Fi(ui, ua, •••, u„) = J^i,
Ft{uu U2, ••., u,) = //,,
I
where J/i, Hs, •••, Hn ft^e constants or functions of certain other variables
3:1, Xj, •.., Xm, which may also occiu: in the functions F,. If the Jacobian
^(Fi, Fj, • . ., F„)/D(ui, Ua, • •, u^) vanishes identically, there exist between
the n fimctions F, a certain number, say n — ilr, of distinct relations of the form
Ft+i = ni(Fi, • . ., Ft), . . . , F, = n„_*(Fi, . , Ft).
68
FUNCTIONAL RELATIONS
[II, § 29
In order that the eqaations (10) be compatible, it is evidently necessary that
Ht^i = ni(£ri, . . ., jJt), . . ., JET, = nn-t(Hi, • • • , ff*),
and, if this be true, the n equations (10) reduce to k distinct equations. We
have then the same cases as in the discussion of a system of linear equations.
29. Another property of the Jacobian. The Jacobiau of a system of n
functions of n variables possesses properties analogous to those of
the derivative of a function of a single variable. Thus the preceding
theorem may be regarded as a generalization of the theorem of § 8.
The formula for the derivative of a function of a function may be
extended to Jacobians. Let Fj, Fj, • ••, F„ be a system of n func-
tions of the variables Wi, Wj, •••, u„, and let us suppose that Wj, U2,
.••, tt, themselves are functions of the n independent variables a^i,
jBg, • • ., X,. Then the formula
D(F,, F„ ■■, f\) __ D(Fiy Fa, >.., F,) D(uu u^,"-, u,)
/)(xi, x,, ..., iTj D{u^,u^y"-,u^) D(xi, Xa, ...,x„)
follows at once from the rule for the multiplication of determinants
and the formula for the derivative of a composite function. For,
let us write down the two functional determinants
dui
dFi
dx.
dx.
dF^ .
J
dui
dx„
where the rows and the columns in the second have been inter-
changed. The first element of the product is equal to
dFi dui dFi dut
dui dxi du^ dxi
du^ dxi
that is, to dFi/dxiy and similarly for the other elements.
•0. BMdaas. Let /(x, y, 2) be a function of the three variables x, y, «. Then
the fuiicUooal det«nninant of the three first parUal derivatives df/dx, df/dy,
V/^.
h =
5/ j^f _av
3x* dxdy dxdz
dxdy V dydz
a*/ y/ y/
dxdt dydz dz*
11. §30]
FUNCTIONAL DETERMINANTS
69
is called the Hessian of /(z, y, z). The Heasian of a function of n rariablet Is
defined in like manner, and plays a rdle analogous to that of the second deriTa-
tive of a function of a single variable. We proceed to prove a remarkable
invariant property of this dei«ruiiuaut. Let us suppose the independent Tari>
ables traiuiformed by the linear subetitution
where X, F, Z are the transformed variables, and or, /9, 7, • • • , y^ are constants
such that the determinant of the substitution.
(1»0
a
^
7
a"
^
7'
a"
r
7'
is not zero. This substitution carries the function /(x, y, z) over into a new
function F(A', F, Z) of the three variables X, F, Z. Let // (X, 1', Z) be the
Hessian of this new function. We shall show that we have identically
H{X, r, Z) = A«A(x.y,z),
where 2, y, z are supposed replaced in A(z, y, z) by their expressions from {W).
For we have
/aF SF aF\ T)/££ cF iF\
^_ \dx' dY' izj _ \dx' ar' ez/ d(2, y, z) .
D(X, r, Z) ~ D(x, y, 2) I>(X, r, Z) '
and if we consider tf/dx^ ^/Z^, ^//^«f for a moment, as auxiliary variables,
we may write
eF £F eF^
ax' ar' eZ/ ~ Vex_£y__£2/ D(x, y, z)
D(x,y, z) 'D(X, F, Z)
£r =
\ax' iy' dz) ygx* ay* c^^/
\ax ay az /
But from the relation F(X, F, Z) =/(x, y, 2), we find
3/
ZF a/ . ,a/ . ,
-— = a— ■ha'— -\-a'
dX dx dy
dz
BY ^ dx ^ dy ^ dz
whence
dZ ^ bx dy
tf
<
dF
ax'
dF
dF
dZ
and hence, finally,
\dx dy czj
P
U = Ah
7
D(x, y, 2)
D(x, r, z)
It is clear that this theorem is general.
dz
a" a"
= A«*.
A;
dO FUNCTIONAL RELATIONS [ll, §30
Let vm now consider an application of this property of the Hessian. Let
/(z, y) = ou^ + 3ftx«y + 3cxy2 + dy*
bt a glrwi Wnmry cubic form whose coefficients a, 6, c, d are any constants.
Hmh, neglecting a numerical factor,
I « + 6y *« + ^ I = (ac - 62) x2 + (od - 6c) xy + (M - c2) y2,
l&t + cycx + di/l
and the Heeelan is seen to be a binary quadratic form. First, discarding the
OMe in which the Hessian is a perfect square, we may write it as the product of
two Unew factors :
h = (mx + ny) (pz + qy).
If, DOW, we perform the linear substitution
mx-^ny = X, px + qy = Yj
the fonn/(x, y) goes over into a new form,
F(X, F) = AX» + 3BX2r+ 3 CXY^ + DF«,
wbaee Heeeian is
H{X, T) = {AC-&)X^ + {AD - BC) XY + {BD - C^) Y%
and this must reduce, by the invariant property proved above, to a product of
the form KXY. Hence the coefficients A, B, C, D must satisfy the relations
Bi-AC = 0, BD-C^ = 0.
If one of the two coefficients £, C be different from zero, the other must be so,
and we shall have
C B
BC BC
whence F(X, F), and hence /(x, y)\ will be a perfect cube. Discarding this
parUcnlar case, it is evident that we shall have B= C = 0; and the polynomial
F(X, F) will be of the canonical form
AX* + BY*.
Hanoe the reduction of the form /(z, y) to its canonical form only involves the
efrintloB of an equation of the second degree, obtained by equating the Hessian
of the giren form to zero. The canonical variables X, Y are precisely the two
faetofi of the Heeelan.
It is eesy to see, in like manner, that the form /(x, y) is reducible to the form
AX* -i- BX* F when the Hessian is a perfect square. When the Hessian van-
iehai identically /(z, y) is a perfect cube :
/(x, y) = (crx + /Jy)«.
II, J 31] TRANSFORMATIONS 61
III. TRANSFORMATIONS
It often happens^ in many problems which arise in Mathematical
Analysis, that we are led to change the independent variables. It
tlierefore becomes necessary to be able to express the derivatives
with respect to the old variables in terms of the derivatives with
respect to the new variables. We have already considered a problem
of this kind in the case of inversion. I^et us now consider the
question from a general point of view, and treat those problems
which occur most frequently.
31. Problem I. Let y be a function of the independent variable jc,
and let t be a new independent variable connected with x by the relation
X = <f>(t). It is required to express the successive derivatives ofy with
respect to x in terms of t and the successive derivatives of y with
respect to t.
Let y=f(x) be the given function, and F(t) =/[<^(0] the func-
tion obtained by replacing x by <f>(t) in the given function. By the
rule for the derivative of a function of a function, we find
di/ du ,, ^
whence
dy
— ^^ — ^''*
^'~ 4>\t)~ 4>\t)
This result may be stated as follows : To find the derivative of y
with respect to x, take the derivative of that function with respect to t
and divide it by the derivative of x with respect to t.
The second derivative d^y /dx^ may be found by applying this
rule to the expression just found for the first derivative. We find :
d
d^y ^ dt^^'^ _ y„<t>'(t)-y,4>"(t) ,
dx^ <f>xt) [<^'(or '
and another application of the same rule gives the third derivative
52 FUNCTIONAL RELATIONS LU,§32
or, perfonning the operations indicated,
The remaining derivatives may be calculated in succession by
ropeated applications of the same rule. In general, the nth deriva-
tive of y with respect to x may be expressed in terms of <^'(^), <f>"(t)j
•••> ♦^■^(O' *^^ *^® ^^^^ ^ successive derivatives of y with respect to
t, ' These formulae may be arranged in more symmetrical form.
Denoting the successive differentials of x and y with respect to ^ by
dx, dy, d*x, d^y, •••, d'^x, dry, and the successive derivatives of y
with respect to x by y', y", •••, y^''^ we may write the preceding
formulae in the form
(20)
„ dxd^y — dyd^x
y " dx^
,„ d*ydx^-Sd^ydxd^x-hS dy (d^xY -
- dyd^xdx
^ - dx^
The independent variable t, with respect to which the differentials
on the right-hand sides of these formulae are formed, is entirely
arbitrary ; and we pass from one derivative to the next by the
recurrent formula
^ dx
the second member being regarded as the quotient of two differen-
tials.
32. ApplicationB. These formulae are used in the study of plane
curves, when the coordinates of a point of the curve are expressed in
terms of an auxiliary variable t. "^
aj=/(0> y = <f>(t)-
In order to study this curve in the neighborhood of one of its points
it is necessary to calculate the successive derivatives y', y", -of y
with respect to « at the given point. But the preceding formulae
give us precisely these derivatives, expressed in terms of the succes-
Stire derivatives of the functions f{t) and <^ (t), without the necessity
II, § 32] TRAN8FOR^f ATTON8 Qg
of having recourse to the explicit • a of y as a function of «,
which it might be very ditticult, i> y, to obtain. Thus the
first formula
gives the slope of the tangent. The value of y" occurs in an impor-
tant geometrical concept, the radius of eurvature, which is given by
the formula
|y"l
which we shall derive later. In order to find the value of /?, when
tlie coordinates x and i/ are given as functions of a parameter t, we
need only replace i/' and y" by the preceding expressions, and we
find
\dxd'^y —dyd^x\
where the second member contains only the first and second deriva-
tives of X and y with respect to t.
The following interesting remark is taken from M. Bertrand^a Traiii de
Calcul diff^rentiel et integral (Vol. I, p. 170). Suppose that, in calculating some
geometrical concept allied to a given plane curve whose coordinates x and y are
supposed given in terms of a parameter t, we had obtained the expression
F(x, y, dx, dy, dPx, dl^y, • • , d-x, d-y),
where all the differentials are taken with respect to t Since, by hypothesis,
this concept has a geometrical significance, its value cannot depend upon the
choice of the independent variable t. But, if we take x = ^ we shall have
dx = d£, cPx = dfix = • • = d«x = 0, and the preceding expression becomes
which is the same as the expression we would have obtained by supposing at the
start that the equation of the given curve was solved with respect to y in the
form y = <J>(x). To return from this particular case to the case where the inde-
I^endent variable is arbitrary, we need only replace y', y", • • by their values
from the formulas (20). Performing this substitution in
we should get back to the expression F(x, y, dx, dy, cPx, d«y, • • •) with which
we started. If we do not, we can assert that the result obtained is incorrect.
For example, the expression
dxcPy -f dyd^x
(dx« -f dy")'
64 FUNCTIONAL RELATIONS [ii,§33
CAiuot have any geometrical significance for a plane curve which is independent
of the choice of the independent variable. For, if we set x = t^ this expression
redooM to ^"/(l + y'*)* ; and, replacing y" and y" by their values from (20), we
do not get back to the preceding expression.
33. The formulffl (20) are also used frequently in the study of
differential equations. Suppose, for example, that we wished to
determine all the functions y of the independent variable x, which
satisfy the equation
(21) (i_.^g_.J + „., = o,
where n is a constant. Let us introduce a new independent variable
t, where x = cos t. Then we have
dy _ dt
~r~ — >
ax — sin t
d}y
dx^
dt"^
"dt.
and the equation (21) becomes.
after the substitution,
(22)
ePv
J, + »v - 0.
It is easy to find all the functions of t which satisfy this equation,
for it may be written, after multiplication by 2 dy/dt,
whence
^SS--/i=l[(S)v»v]=o,
where a U an arbitrary constant. Consequently
g = „V^T37,
-r- - - » = 0.
U. § -M] TRANSFORMATIONS 66
The left-hand side U the derivative of arc sin (y /a) — nt. It follows
that this difference must be another arbitrary constant b, whence
y = a8in(n^ 4- b),
which may also be written in the form
1/ = A Binnt -^ B cos nt
Returning to the original variable x, we see that all the functions of
X which satisfy the given equation (21) are given by the formula
y = A sin (n arc cos x)-^ B cos (n arc cos x),
where A and B are two arbitrary constants.
34. Problem II. To evert/ relation between x and y there corresponds,
by means of the transformation x =f{tj u), y =: ifi(^tf w), a relation
between t and u. It is required to express the derivatives of y with
respect to x in terms of tj u, and the derivatives of u with respect to t.
This problem is seen to depend upon the preceding when it is
noticed that the formulae of transformation,
give us the expressions for the original variables x and y as func-
tions of the variable t^ if we imagine that u has been replaced in
these formulae by its value as a function of ^. We need merely
apply the general method, therefore, always regarding x and y as
composite functions of ty and w as an auxiliary function of t. We
find then, first,
dy
dy
dx
d<i> d<i>
dt "^ du
du
dt
dx
= ;
dt
' dt
cf.df
ct "^ du
7Z'
dt
and then
d^y __ d (dy\ dx
'dx^~"dt \dx/ ' 'di '
or, performing the operations indicated,
\H hi dtl\_H^ hi ct dt du\dt ) eudt^J \rt cu dl l[_ £t^ J
\ft "*" cH di)
^ FUNCTIONAL RELATIONS [II, §35
In ^neral, the nth derivative y<"> is expressible in terms of t, u, and
the derivatives du/dt, d^u/d^, • -, d^u/dt\
Suppose, for instance, that the equation of a curve be given in
polar codrdinates p =/(o>). The formulae for the rectangular coor-
dinates of a point are then the following :
X = p cos 0), y = p sin <i).
Let p', p", . • • be the successive derivatives of p with respect to w,
considered as the independent variable. From the preceding formulae
we find
dx = cos Hi dp — p sin o) c^w,
dtj = 8ija.n)dp + p cos w e?o»,
d^x = coscD d^p — 2 sino) dm dp — p cos to c?o)^,
d^y =^ sin a> cPp + 2 cos (o dot dp — p sin w c?<u^,
whence
dx^ + d}/ = dp^-hp^dio^,
dx d^y — dij d^x = 2 d(o dp^ — p dto d^ p -h P^ d<oK
The expression found above for the radius of curvature becomes
R = ±
p^4-2p'^-pp'
35. Transformations of plane curves. Let us suppose that to every
point m of a plane we make another point M of the same plane cor-
respond by some known construction. If we denote the coordinates
of the point m by (x, y) and those of M by (X, F), there will exist,
in general, two relations between these coordinates of the form
(23) X=f{x,y), Y=<t>(x,y).
These formulae define a point transformation of which numerous
examples arise in Geometry, such as projective transformations, the
transformation of reciprocal radii, etc. When the point m describes
a curve c, the corresponding point M describes another curve C, whose
properties may be deduced from those of the curve c and from the
nature of the transformation employed. Let y\ y", -be the suc-
oeisive derivatives of y with respect to x, and F', F", • • • the succes-
•ive derivatives of Y with respect to X. To study the curve C it
is necessary to be able to express F', F", -in terms of x, y, y\
y"t"'» This is pre<u8ely the problem which we have just discussed ;
and we find
11. §36]
TRANSFORMATIONS
67
dY
dx
dX
dx
dY'
dx_
dX
dx
dx^ dy^
and so forth. It is seen that F' depends only on «, y, y\ Hence^
if the transformation (23) be applied to two curves r, c\ which are
tangent at the point (a^, y), the transformed curves C, C will also
be tangent at the corresponding point {X^ Y). This remark enables
us to replace the curve c by any other curve which is tangent to it
in questions which involve only the tangent to the transformed
curve C,
Let us consider, for example, the transformation defined by the
formula3
h'^x __ hhj
X =
:' + y'
Y =
-fy'
which is the transformation of reciprocal radii, or inversion^ with
the origin as pole. Let vi be a point of a curve c and M the cor-
responding point of the curve C. In
order to find the tangent to this curve
C we need only apply the result of
ordinary Geometry, that an inversion
carries a straight line into a circle
through the pole.
Let us replace the curve c by its
tangent mt. The inverse of mt is a
circle through the two points M and O,
whose center lies on the perpendicular
Ot let fall from the origin upon vit. The tangent MT to this circle
is perpendicular to AM, and the angles Mmt and mMT are equal,
since each is the complement of the angle viOL The tangents mt
and MT are therefore antiparallel with respect to the radius vector.
36. Contact transformations. The preceding transformations are
not the most general transformations which carry two tangent
curves into two other tangent curves. Let us suppose that a point
M is determined from each point m of a curve c by a construction
Fio. 5
68 FUNCTIONAL RELATIONS [II, §36
which depends not only upon the point m, but also upon the tangent
to the curve c at this point. The formulaB which define the trans-
formation are then of the form
(24) X =/(x, y, y'), Y = 4> (x, y, y') ;
and the slope Y' of the tangent to the transformed curve is given
by the formula
dY _ dx^ dy^^dy'^
dx^ dy^ ^ dy'^
In general, F' depends on the four variables x, y, y\ y"; and if we
apply the transformation (24) to two curves c, c' which are tangent
at a point (x, y), the transformed curves C, C will have a point
(JT, F) in common, but they will not be tangent, in general, unless
y" happens to have the same value for each of the curves c and c'.
In order that the two curves C and C should always be tangent, it
is necessary and sufficient that F' should not depend on y"; that is,
that the two functions /(x, y, y') and <^ (x, y, y ') should satisfy the
condition
a//a^ a^ \ a^/V a/ A.
dy\dx^ dy^j dy\dx^dyy )
In case this condition is satisfied, the transformation is called a
contact transformation. It is clear that a point transformation is a
particular case of a contact transformation.*
Let us consider, for example, Legendre's transformation, in which
the point 3^, which corresponds to a point (x, y) of a curve c, is given
by the equations
a: = y', F = xy' - y ;
from which we find
■>
dX
y" ~
■ .«.,
which shows that the transformation is
a contact transformation.
In like manner we find
F"
dY'
dX
dx
y"dx
1
ylii
dY"
dX
y-'
*J*T**'T *"** Ampfcw gave many examples of contact transformations. Sophus
see in particular his Geornet
Vorledungen iiber Dynamik.
U«4«Taloped the general theory in various works; see in particular his Geojuetrie
d$r Btrahrunattran^onnatioMn. See also Jacobi
II. $37] TRANSFORMATIONS 69
and 80 fortli. From the preceding formulae it follows that
which shows that the transformation is involutory.* All these prop-
erties are explained by the remark that the point whose coordinates
are -Y = y', Y = xi/' — y is the pole of the tangent to the curve c at
tlie point (x, y) with respect to the parabola x* — 2 y = 0. But, in
general, if 3/ denote the pole of the tangent at w to a curve c with
respect to a directing conic 2, then the locus of the point M is a
curve C whose tangent at M is precisely the polar of the point m
with respect to 2. The relation between the two curves c and C is
therefore a reciprocal one ; and, further, if we replace the curve c by
another curve c', tangent to c at the point m, the reciprocal curve C
will l)e tangent to the curve C at the point M.
Pedal curve.s. If, from a fixed point in the plane of a curve c, a perpen-
dicular OM be let fall upon the tangent to tlie curve at the point m, the locus of
the foot 3/ of this perpendicular is a curve C, which is called the pedal of the
given curve. It would be easy to obtain, by a direct calculation, the coordinates
of the point M, and to show that the trans-
formation thus defined is a contact transfor-
mation, but it is simpler to proceed as follows.
I^et us consider a circle y of radius H, de-
scribed about the point O as center; andlet»ni
be a point on 03/ such that Omi x 0M= RK
The point mi is the pole of the tangent mt
with respect to the circle; and hence the
transformation which carries c into C is the
result of a transformation of reciprocal po-
lars, followed. by an inversion. "When the
point /n describes the curve c, the point mi,
the pole of mt, describes a curve Ci tangent Fio. 6
to the polar of the point m with respect to
the circle 7, that is, tangent to the straight line mi<i, a perpendicular let fall
from mi upon Ojh. The tangent 3f Tto the curve C and the tangent mi^i to the
curve Ci make equal angles with the radius vector OniiM. Hence, if we draw
the normal MA, the angles AMO and A OM are equal, since they are the comple-
ments of equal angles, and the point A is the middle point of the line Om. It
follows that the normal to the pedal is found by joining the point 3f to the center
of the line Om.
37. Projective transformations . Every function y which satisfies the equation
V" = is a linear function of x, and conversely. But, if we subject x and y to
the projective transformation
*That is, two suocessive applications of the transformation lead us back to the
original coordinates. — Trans.
70 FUNCTIONAL RELATIONS [II, $38
aX + 6r+c a' X -\- h' Y ■\- cT
z =
a^'X-k-lf'Y-^cf' a"X-\-h"Y + &'
% straight line goes over into a straight line. Hence the equation y" = should
d^T/dX* = 0. In order to verify this we will first remark that the
projective transformation may be resolved into a sequence of particular
transformations of simple form. If the two coefficients a^' and y are not both
lero, we will set Xi = a"X -\- h" Y + c" \ and since we cannot have at the same
time aft" - ha" = and a'h" — h'a.*' = 0, we will also set Fi = a'X + 6' F + c',
on the supposition that a'h" — h' a" is not zero. The preceding formulas may
then be written, replacing X and Y by their values in terms of Xx and Fi, in
the form
^1 Xx ^ Xx Xx
It follows that the general projective transformation can be reduced to a
succession of integral transformations of the form
x = aX+6F+c, y = a'X + 6'F + c',
combined with the particular transformation
1
F
this latter transformation
, we find
dx
XY'-
F -1
* X2 ~
F - XY\
and
y" = ^ = - XY'\- X^\ = X8 F''.
dx '
Likewise, performmg an integral projective transformation, we have
dx a + &F'
y// _ ^' _ {ab'-ba')Y'^
dx {a-\-bYy
In each case the equation y" = goes over into F" = 0.
We shall now consider functions of several independent variables, and, for
dtflniteoess, we shall give the argument for a function of two variables.
38. Problem in. Let a> =/(x, y) be a function of the two independ-
ent variables x and y, and let u and v be two new variables connected
mth the old ones by the relations
ap = <^(tt, v), y = \ff(u, v).
h is required to express the partial deHvatiues of <u tvith respect to the
variables xandy in terms of w, v, and the partial derivatives of <u with
'"* to u and v.
n, J 38] TRANSFORMATIONS 71
Let a» = F(w, v) be the function which results from /(x, y) by the
substitution. Then the rule for the differentiation of composite
functions gives
dm dm d<f> dot dd/
ss — — ■ -^ ^f
du dx du dy du
dm dm dtf> dm dtp
dv dx dv dy dv
whence we may find dm/dx and dm/dy\ for, if the determinant
l){<^, \l/)/D(u, v) vanished, the change of variables performed
wuuld have no meaning. Hence we obtain the equations
(25)
dm dm dm
dx du du
dm ^ dm . ^ dm
cy cu dv
where A^ Bj C, D are determinate functions of u and v; and these
formulae solve the problem for derivatives of the first order. They
show that the derivative of a function with respect to x is the sum of
the two products formed by multiplying the two derivatives with respect
to u and v by A and B, respectively. The derivative with respect to
y is obtained in like manner, using C and D instead of A and B,
respectively. In order to calculate the second derivatives we need
only apply to the first derivatives the rule expressed by the preced-
ing formulae ; doing so, we find
d^m d (dm\ ^ ( A ^^ _t_ c>^*^\
dx^ dx\dx/ dx\ du dv)
d I dm d m\ d I dm dm\
du\ du dv I dv\ du dv)
or, performing the operations indicated,
dx^ \ du^ du dv du du du dv )
\ dudv dv^ dv du dv dv)
and we could find d*m/dxdy, d*m/dy^ and the following derivatives
in like manner. In all differentiations which are to be carried out
we need only replace the operations d/dx and d/dyhy the operations
c? a d d
du dv du dv
72 FUNCTIONAL RELATIONS [II, §38
respectively. Hence everything depends upon the calculation of the
ooefficients A, By C, D.
Example I. Let us consider the equation
/90\ o — - + 2 — — - + c -— = 0,
where the coefficient* a, 6, c are constants ; and let us try to reduce this equa-
tion to as simple a form as possible. We observe first that if a = c = 0, it would
be superfluous to try to simplify the equation. We may then suppose that c,
for example, does not vanish. Let us take two new independent variables u
and 0, defined by the equations
M = X 4- ay, v = z-\- py,
where a and /3 are constants. Then we have
dx '
du du
~ du dv
d(o
du , ^du
and hence, in this case, A = B-\, C = a, D = p. The general formulae then
fdve
S^u _d^u ^ d-2u d^u
a»2 "" du'^ dudv dv^
^u ^20,,, , ^, d'^u , d'^u
dxdy du"^ ^ 'dudv dv^
dy^ aw* "^dudv dv^
and the given equation becomes
(a + 26a + ca«)^ + 2 [a + 6(a + /3) + ca/3]£^ + (a -i- 26^ + c^)^ = 0.
It remains to distinguish several cases.
First case. Let l^ - ac>0. Taking for a and /3 the two roots of the equation
a + 2 6r + cr* = 0, the given equation takes the simple form
Since this may be written
= 0.
dudv
dv\du/
we Me that du/du must be a function of the single variable, u, 8ay/(u). Let
P(u) denote a function of u such that F'(u) =/(u). Then, since the derivative
of •» — F(u) with respect to u is zero, this difference must be independent of m,
and, accordingly, u = F{u) -f ♦ (v). The converse is apparent. Returning to
the varlablea x and y, it follows that all the functions u which satisfy the equation
(90) are of the form
w = F(x + ai/) + «l>(x + /3y),
Il,§a«J TRANSFORMATIONS 78
where F and ^ are arbitrary functioua. For example, the geiMral Inlegral of
the equatiun
— = a* — f
which occurs iu the theory of the stretched string, Is
«=/(« + «y) + (« - ay).
Second case. Let b^ — ac = 0. Taking a equal to the double root of the equa-
tion a + 26r + cr3 = 0, and /3 some other number, the coefficient of d^ta/cutv
becomes zero, for it is equal to a + 6a + /3 (6 + ca). Hence the given equation
retluces to d^u/tv'^ = 0. It is evident that « must be a linear function of v,
u = vf(u) + (w), where /(u) and (u) are arbitrary functions. Returning to
the variables z and y, the expression for w becomes
« = (X + /3y)/(x + ay) + 0(x + ay),
which may be written
w = [x + «y + 09 - «)y]/(-c + ay) 4- 0(x + ay),
or, finally,
« = yF{x + ay) + «f>(x + ay).
Third case. If 6=* - ac < 0, the preceding transformation cannot be applied
without tlie introduction of imaginary variables. The quantities a and /3 may
then be determined by the equations
o + 2 6a-fca« = a + 2 6/3 + c/3»,
a + 6(a + /3) + ca/3 = 0,
which give
26 ^ 26«-ac
a + p = , a/5=
c c*
The equation of the second degree,
. 26 ^ 26«-ac ^
r« + — r4- — = 0,
c c*
whose roots are a and /3, has, in fact, real roots. The given equation then
becomes
Aw = — - + — - = 0.
au« at*
This equation Aw = 0, which is known as Laplace's Equation, is of fundamental
importance iu many branches of mathematics and mathematical physics.
Example II. Let us see what form the preceding equation assumes when we
set X = /} COS0, y = p sin0. For the first derivatives we find
dw dw ^ . 5« , ^
— = —0080 + — sin 0,
dp dx cV
- = --(,»ln«+-,co.#.
74 FUNCTIONAL RELATIONS [II, §39
or» solving for a«*/ac and e«/ay,
Su du 8in0 dbt
— = CO80 »
dx dp p d<f>
dta du COS0 d<a
— =8in0 1 — •
dy dp p o<p
?i = CO.*- (co.* -- — -)-— ^ (cos* ^ - — -j
a«« 8in«0 52« 2 8in0co80 d'^u 2,&m<t>cos<t> d(a sm^^ 8fa>
and the expreasion for d^u/dy^iB analogous to this. Adding the two, we find
^"^ay2~ap^"^p2^ pap'
39. Another method. The preceding method is the most practical
when the function whose partial derivatives are sought is unknown.
But in certain cases it is more advantageous to use the following
method.
Let z =f(xj ^) be a function of the two independent variables x
and y. If x, y, and z are supposed expressed in terms of two aux-
iliary variables u and v, the total differentials dx, dy, dz satisfy the
relation
which is equivalent to the two distinct equations
du dx du dy du
^ = £/*?^ + ?/?i^,
dv dx dv dy dv
whence df/dx and df/dy may be found as functions of u, v, dz/duy
dg/dvy as in the preceding method. But to find the succeeding
derivatives we will continue to apply the same rule. Thus, to find
^f/dz* and ^fldxdy, we start with the identity
<i)'
iff a»/
which is equivalent to the two equations
dx'^ du dx dy du
II, §39] TRANSFORMATIONS 76
(^)_
ayax ay dy
dv do^ bv dx dy dv
where it is supposed that dfjdx has been replaced by its value cal-
culated above. Likewise, we should find the values of d^f/dx dy and
d*f/dy^ by starting with the identity
{©■
ay ^^ ay ^
dx dy dy*
The work may be checked by the fact that the two values of
d*f/dzdy found must agree. Derivatives of higher order may be
calculated in like manner.
Apjylication to surfaces. The preceding method is used in the study
of surfaces. Suppose that the coordinates of a point of a surface ^'
are given as functions of two variable parameters u and v by means
of the formulae
(27) X = /(m, v), y = <\> (w, v), z = ^(u,v).
The equation of the surface may be found by eliminating the vari-
ables u and V between the three equations (27); but we may also
study the properties of the surface S directly from these equations
themselves, without carrying out the elimination, which might be
practically impossible. It should be noticed that the three Jacobians
D{f.<\>) D{4>^^) D(f, «A)
D(uy v) ' D(Uy v) * D{u, V)
cannot all vanish identically, for then the elimination of u and v
would lead to two distinct relations between x, y, z, and the point
whose coordinates are {x, y, z) would map out a curve, and not a sur-
face. Let us suppose, for definiteness, that the first of these does not
vanish : /)(/, <fi)/D(u, r) ^ 0. Then the first two of equations (27)
may be solved for u and u, and the substitution of these values in the
third would give the equation of the surface in the form z = F(Xj y).
In order to study this surface in the neighborhood of a point we need
to know the partial derivatives p, q, r, s,t, • • of this function F(j*, y)
in terms of the parameters u and v. The first derivatives p and q
are given by the equation
dz = pdx + q dyj
which is equivalent to the two equations
75 FUNCTIONAL RELATIONS [n,§40
(28)
dd, df. d4>
from which p and q may be found. The equation of the tangent
plane is found by substituting these values of i? and s' in the equation
Z-z=p(X-x)-¥qiY-y)y
and doing so we find the equation
The equations (28) have a geometrical meaning which is easily
remembered. They express the fact that the tangent plane to the
surface contains the tangents to those two curves on the surface which
are obtained by keeping v constant while u varies, and vice versa*
Having found i? and q, p =/i(m, v), q =A(u, v), we may proceed
to find r, «, t by means of the equations
dp = rdx + sd]/j
dq = sdx -^ tdy,
each of which is equivalent to two equations ; and so forth.
40. Problem IV. To every relation between x, y, z there corresponds
hy meant of the equations
(30) X =f{u, v, w), y = <f> (u, V, w), z = if;(u, v, w),
a new relation between w, v, w. It is required to express the partial
derivatives of z with respect to the variables x and y in terms of u, v, Wj
and the partial derivatives of w with respect to the variables u and v.
This problem can be made to depend upon the preceding. For,
if we suppose that w has been replaced in the formulae (30) by a
function of u and v, we have a;, y, z expressed as functions of the
•Th« «qamtion of the tangent plane may also be found directly. Every curve on
tb* wutmM is defined by a relation between u and v, say v = n (u) ; and the equations
dl tlM toogent to ibis curve are
Y-
tu tv du dv du 8v
Th* idiiuliintloii iif lV(u\ jM«di to the equation (29) of the tangent plane.
II, §41]
TRANSFORMATIONS
77
two paxameters u and v; and we need only follow the preceding
method, considering f,<f>yilfas composite functions of u and v, and
t^; as an auxiliary function of u and v. In order to calculate the
first derivatives p and q, for instance, we have the two equations
^.^^^ (^,^^\, (^i^^\
du dw du \du dw du ) ^\du dw duj*
diff
dw dv \dv dw dv / ^\dv dw dv/
The succeeding derivatives may be calculated in a similar manner.
In geometrical language the above problem may be stated as fol-
lows : To every point m of space, whose coordinates are (x, y, «),
there corresponds, by a given construction, another point 3/, whose
coordinates are A", Y, Z. When the point m maps out a surface 5,
the point M maps out another surface 2, whose properties it is pro-
posed to deduce from those of the given surface .S'.
The formulae which define the transformation are of the form
Let
^ = /(«» y» «)» Y^^ («, y, 2), Z = ^ (x, y, «).
be the equations of the two surfaces S and 2, respectively. The
problem is to express the partial derivatives P, Q, Rj S^T, -" of the
function *(A', Y) in terms of x, y, z and the partial derivatives
Pi ?> ^i s, t, • ■ of the function F(x, y). But this is precisely the
above problem, except for the notation.
The first "derivatives P and Q depend only on x, y, z, p, q; and
hence the transformation carries tangent surfaces into tangent sur-
faces. But this is not the most general transformation which enjoys
this property, as we shall see in the following example.
41. Legendre's transformation. Let z =/(x, y) be the equation of
a surface .S', and let any point m (x, y, z) of this surface be carried
into a point 3/, whose coordinates are A', 1', Z, by the transformation
X=p, F = 7, Z ^px-^qy — z.
Let Z = <I>(A', Y) be the equation of the surface 2 described by the
point M. If we imagine z, p, q replaced by /, df/dxy df/dy^ respec*
tively, we have the three co6rdinates of the point 3/ expressed as
functions of the two independent variables x and y.
78 FUNCTIONAL RELATIONS [II, §41
Lot P, Q, R, Sy T denote the partial derivatives of the function
♦(X, Y). Then the relation
rfZ= PdX-\- QdY
becomes , _
pdx-{-qdi/ + xdp + ydq — dz = Pdp-\'Qdq,
or
xdp-\-ydq = Pdp+ Qdq.
Let U8 suppose that j9 and q, for the surface S, are not functions of each
other, in which case there exists no identity of the form \dp-\-fi.dq — Oj
unless X = /i = 0. Then, from the preceding equation, it follows that
p = x, Q^y-
In order to find R, S, T we may start with the analogous relations
dP = RdX-{- SdY,
dQ = SdX-{-TdY,
which, when X, F, P, Q are replaced by their values, become
dx = R(rdx -\- sdy) + S{sdx + tdy),
dy = S (rdx + s dy) -t T(s dx + tdy)\
whence
Rr^-Ss=l, Rs + St = Oy
Sr-{-Ts = Oy Ss-\-Tt = ly
and consequently
^^^iiZTi^' ^^H^^r?' ^=;^zr^-
From the preceding formulae we find, conversely,
x = P, y=Q, z = PX+QY-Z, p = X, q = Y,
T - S R
t = •— -n -:;>
RT-S^ RT-S^ RT-S^
which proves that the transformation is involutory. Moreover, it
is a contact transformation, since X, Y, Z, P, Q depend only on x,
y» «» Pt 9' These properties become self-explanatory, if we notice
that the formulae define a transformation of reciprocal polars with
respect to the paraboloid
NaU. The expressions for R^ S, T become infinite, if the relation
i< — *• = holds at every point of the surface S. In this case the
point M describes a curve, and not a surface, for we have
II, § 42] TRANSFORMATIONS 79
D(x, y) D(x, y)
and likewise
DiX,Z) D(p,px + qy-z)
D(x,y) D(x,y) '^^"^ '^-'^-
This is precisely the case which we had not considered.
42. Ampere's transfonnatioD. Retaining the notation of the preceding article,
let lis consider the transformation
X = X, r = g, Z-qy-z,
The relation
dZ-TdX-^ QdY
becomes
qdy + ydq - dz = Pdx + Qdq,
or
ydq —pdx = Pdx-\- Qdq.
Hence
and conversely we find
x = X, y=Q, z=QT-Z, p = - P, q = T.
It follows that this transformation also is an involutory contact transformation.
The relation
dP = RdX-{-8dT
next becomes
-rdx -8dy = Rdz-\- 8{8dx -{-tdy)',
that is,
B + 5« = - r, 8t= - 8,
whence
t t
Starting with the relation dQ = SdX + TriF, we find, in like manner,
t
As an application of these formulae, let us try to find all the functions /(z, y)
which satisfy the equation rt — s^ = 0. Let S be the surface represented by the
equation z =/(x, y), 2 the transtonned surface, and Z = *(X, Y) the equation
of 2. From the formulas for /2 it is clear that we must have
and 4> must be a linear function of X :
z = X0(r) + ^(r),
where and \f/ are arbitrary functions of T. It follows th»t
P = <f>{Y), Q = X0'(r) + v^'(F);
FUNCTIONAL RELATIONS [U, §43
oonrersely, the coordinates (x, y, z) of a point of the surface 8 are given
M fonctioDB of the two variables X and Y by the formulae
« = X. y = x^'(r) + rm, z = r[X0'(r) + \^'(F)] - x<f>{T) - ^ (F).
TIm eqiiAtion of the surface may be obtained by eliminating X and Y ; or, what
amounts to the Sftine thing, by eliminating a between the equations
z = ay-x<p{a)-yl/{a),
= y -X <p'{a) - yp^{a).
The first of these equations represents a moving plane which depends upon the
parameter a, while the second is found by differentiating the first with respect
to this parameter. The surfaces defined by the two equations are the so-called
dtMJopabU surfaces, which we shall study later.
4S. The potential equation in curvilinear coordinates. The calculation to which
a change of variable leads may be simplified in very many cases by various
derioes. We shall take as an example the potential equation in orthogonal
earviUnear coordinates.* Let
F (X, y, z) = p,
Fi{x, y, z) = pi,
■P2(x, y, Z) = p2,
be the equations of three families of surfaces which form a triply orthogonal
system, such that any two surfaces belonging to two different families intersect
at right angles. Solving these equations for x, y, z as functions of the parame-
iers p, ^1, ps, we obtain equations of the form
« = <t> {p, Pu Pi)i
(SI)
{« = 0>, Pi, Pih
y = 0i(/>, pi, P2),
2 = 02(/), Pi, Pa);
and we may take p, pi , ps as a system of orthogonal curvilinear coordinates.
Since the three given surfaces are orthogonal, the tangents to their curves of
interseotion must form a trirectangular trihedron. It follows that the equations
most be aatisfled where the symbol S indicates that we are to replace by 0i,
then by ^, and add. These conditions for orthogonalism may be written in the
loUowing form, which is equivalent to the above :
m
5p£pi_^apapiap£pi_
dx dx dy dy dz dz ~ '
KOZ ax dx dx
dtg^Ul^\T'. JJl*'^'*'^*' cur^ign^B. See also Bertrand, TrcdU de CcUcul
u. § 4;ij
TKA2sbFOUM Allocs
81
Let us then see what form the potential equation
in the variables />, pi, ps . First of all, we find
dx dp dx dpi dx dpt dx
and then
^V _ !^V (Spy , 2 g*^ ^P dpi dV ^p
dx^ d(^\dxj dpdpi dx dx dp d^
dVd»pi
^ ^/apiy ^ 2^F_ apt ap, ^ ar
apj \dx/ dpi dpt dx dx dpi
.^(dptV
dfi \dx)
dpdpt dx dx dp% dx^
Adding the three analogous equations, the terms containing derivatives of the
second order like c^ V / dp cpi fall out, by reason of the relations (33), and we have
(34)
axa
cy^ dz^ dp^ cp\ dp\
+ Aa(p) ^' + A,(pi) ^' + A,(p,) 1^,
ap api dpt
where Ai and Aa denote Lami*B differential parameters :
ay2 cz*
The differential parameters of the first order Ai (p), Ai (pi), Ai (ps) are easily
calculated. From the equations (31) we have
a0 dp a0 dp\ a0 apa
ap ax api ax aps ax
a0i dp^ d<t>i dpi a 01 a Pa
dp dx dpi ex dpt dx
d4>t dp a0t api a^j ap»
ap a* api ax apj ax
whence, multiplying by
a a 01 a 01
ap dp dp
dp
dx
« respectively, and adding, we find
d<f>
dp
m-<-sy<^}
Then, calculating dp/c'y and dp/dz in like manner, it is easy to see that
1
W <%)■<£)■ -777^
m'^fY^m
Y
S2 FUNCTIONAL RELATIONS [n,§43
Let OB now Mi
"sm- -'S{^r- -sm
where the symbol ^ Indicatee, as before, that we are to replace by 0i, then
by ^, and add. Then the preceding equation and the two analogous equations
may be written
AiO>) = -^' AiO>i) = — , Ai(p2) = — •
H *^*' Hi '^"' H.
Lam4 obtained the expressions for Aj (p), A2 (pi), A2 (pa) as functions of p, pi,
^ by a rather long calculation, which we may condense in the following form.
In the idenUty (34)
AiF=-— - + — — y + — -— y + Aap)— + A2(pi) — - + A2(P2) — >
H ap* Hi df^ Hi ap2 ap api ap2
let us set succeasively F = x, F = y, F = z. This gives the three equations
1 a»0 1 a«0 1 av . . , V S0 a^ a0 _
H V ^^^ ^:s^ ""^"^'^^ +^^^^>^, +^^^^^>^2 =''
n- TT + W" TT + ^ VF + ^2 (p) -— • + A2(pi) -— + A2 (P2) -^- = 0,
B ap* Hi dpi Ht dp2 ^P dpi dpi
1 a«^ 1 a»0, 1 a202 . . , . a02 , . , . a02 , . . . a02 ^
w HT "•■ IF TF + w Ti" + ^a(^) ^- + ^2(pi) ^— + A2(p2) ^ = 0,
H dpi* Hi dp^ Hi dpi dp dpi dpi
which we need only solve for Aj (p), A2 (pi), A2 (pi). For instance, multiplying
by d^/dpy d^/dp^ dipi/dpy respectively, and adding, we find
A-Z-^Wj. ^ e^^^^x ^ nd<t>d^<t>, 1 nd<f>d2<f>_
MbvMTer, we have
oa0 a20 lag
^ ap ap2 ~ 2 ap *
and differentiating the first of equations (32) with respect to pi, we find
oa0a«0___ nd<p 8^<f> _ 1 dHi
*^ dp^ ^ dpidpdpi~ 2 dp '
In like manner we have
*^ ap dpi " 2 ap '
MdMonqontiy
A.(#)---L?f 4.-_^_«^t . 1 dHt_ 1 ar / g \n
iHtf^^'^nni'W^^Wm'di'-'^VpV'^KWHjJ'
II. EX8.]
Setting
thiB formula becomes
"=r«'
£X£RCISES
^' = M'
88
^• = i^'
and in like manner we find
Hence the formula (84) finally becomes
ap2 ap \ ^AiV 2/>
. .op^r a /, A, \ar"
(86)
ax« ai/a az2
or, in condensed form,
Ldp \hihi dp I dpi \hht dpi I dpi \hhy dpi/J
Let us apply this formula to polar coordinates. The formulaB of transforma-
tion are
X = psin(?cos0, y = psindeiiKp, z = pcoB$,
where 6 and <f> replace pi and po, and the coeflRcients h. h^. h^ have the following
Talues:
A=l, hi=\ /i, = -i_.
p p sin 9
Hence the general formula becomes
AaF= — (p^sin^ ) + — (smtf — ) + — ( I h
P^sin e\_dp\ dp dd\ de d<f> Vsin e d4> J
or, expanding,
AaF
dp^ p2 d0^ p«8ln«tf d<t^ ' p dp
a*F 2 aF cottf aF
which is susceptible of direct verification.
EXERCISES
1. Setting u = x« + y2 + 2«, tj = X 4- y + z, w = xy + yz + «x, the functional
determinant D(u, t>, w)/D(x, y. z^ vanishes identically. Find the relation which
exists between u, o, to.
OtMTolixe the problem.
OA FUNCTIONAL RELATIONS [H. Em.
VT^
S. Using the notation
Xi = cos 01,
Xt = 8in0icos02,
Xt = sin 01 sin ^a cos 0$,
»
x« = sin 01 sin 02 • • • sin 0„ _ 1 cos 0„,
•how that
Djxu x«« -t »«) = ( _ i)n sin" 01 sin" - 1 02 sin" - 208 . • • sin2 0»_i sin 0„.
D(0i.0sf •,0»)
4. Prove directly that the function z = F(x, y) defined by the two equations
z = ax-h yf{a) + (a),
= X + yria) + 0'(a),
where a is an auxiliary variable, satisfies the equation rt - s"^ = 0, where f{a)
Aod 0(<r) are arbitrary functions.
5. Show in like manner that any implicit function z = F{x, y) defined by
an equation of the form
y =X0(z) + t/'(z),
where ^ (z) and ^ (z) are arbitrary functions, satisfies the equation
rg2 _ 2i)5S + tp2 = 0.
6. Prove that the function z = F{x, y) defined by the two equations
z 0'(a) = [y - (or)] 2, (X + a) 0'(a) = y - (or),
where a la an auxiliary variable and {a) an arbitrary function, satisfies the
equation pq = t.
7. Prove that the function « = F(x, y) defined by the two equations
[t - 0(a)]« = x«(y« - a«), [2-0 (a)] 0'(a) = ax^
■■tliflei in like manner the equation pq = xy.
fl». Lafraage's formula. Let y be an implicit function of the two variables
X Mid a, defined by the relation y = or + x0(y); and let u =f{y) be any func-
tion of y whatever. Show that, In general,
ftp* do— iL (fa J
[Laplace.]
II, ExB.J EXERCISES 85
Note. The proof Ib based upon the two formal»
where it !■ any function of y whatever, and F(u) is an arbitrary function of 11.
It la shown that if the formula hulds for any value of n, it must hold for the
value n + 1.
Setting z = 0, y reduces to a and u to /(a); and the nth derivative of u with
respect to x becomes
(a=.^.[*<''>"^'<^>]-
9. If X =/(u, 0), y = (u, v) are two functions which satisfy the equations
show that the following equation is satisfied identically :
10. If the function F(x, y, z) satisfies the equation
show that the function
I
exa aya dz^
r \ r^ r^ r*/
satisfies the same equation, where k is a constant and r^ = x^ + y^ + z>.
[Lord Kelvin.]
11. If F(x, y, 2) and Fi(z, y, z) are two solutions of the eijuation AjF = 0,
show that the function
U = F(x, y, 2) + (x« + y2 + z^) Fi(x, y, «)
satisfies the equation
12. What form does the equation
(X - x»)y"+ (1 - 3x2)y'- xy =
assume when we make the transformation x = Vl —t^?
13. What form does the equation
?f + 2xy«?^ 4- 2(y - y«)?^ + x^y^z =
dx^ dx cy
assume when we make the transformation x = ut>, y = l/o ?
14*. I^t (xi, xt, • • • , Xn ; ui, Us, • • •, Um) be a function of the 2n independent
variables xi, x^, • • • , x^, ui, us, •••,!««, homogeneous and of the second degree
with respect to the variables Wi, mj, • • •, u„. If we set
M FUNCTIONAL RELATIONS [U, Exs.
and then take pi , Pj, • • • , P» as independent variables in the place of Wi, 1*2, • • • , Wn,
the fancUoD ^ goes over into a function of the form
D«riTe the formuls : ., ^^
dpk ^k OXk
16. Let JV be the point of intersection of a fixed plane P with the normal MI^
erected at any point Jf of a given surface S. Lay off on the perpendicular to the
plane P at the point N a length Nm = NM. Find the tangent plane to the
surface described by the point m, as 3f describes the surface S.
The preceding transformation is a contact transformation. Study the inverse
transformation.
16. Starting from each point of a given surface S, lay off on the normal to
the surface a constant length I. Find the tangent plane to the surface S {tlie
parallel aurface) which is the locus of the end points.
Solve the analogous problem for a plane curve.
17*. Given a surface S and a fixed point O ; join the point O to any point M of
the surface 8, and pass a plane OMN through OM and the normal MN to the
surface S at the point Jf. In this plane OMN draw through the point a per-
pendicular to the line OM, and lay off on it a length OP = OM. The point P
describes a surface S, which is called the apsidal surface to the given surface S.
Find the tangent plane to this surface.
The transformation is a contact transformation, and the relation between the
surfaces 8 and 2) is a reciprocal one. When the given surface S is an ellipsoid
and the point is its center, the surface Z is FresnePs wave surface.
18*. Halphen'8 differential invariants. Show that the differential equation
\dxV dx6 d«2 dx« dx* \dx»/
remaina unchanged when the variables cc, y undergo any projective transfor-
mation (S 87).
19. If in the expression Pdx -f Qdy -\- Rdz, where P, Q, R are any functions
of X, y, z, we set
x=/(u, t), u>), 2/ = (u, t), ry) , z = rp {u, v, w) ,
whan tt, V| 10 are new variables, it goes over into an expression of the form
Pidu+ Qidt» + Bidiy,
Pit Qi* J^i are functions of u, t), w. Show that the following equation is
identioally:
D{u, t>, to)
II. Ex«.] EXKRCI8E8 87
when
''■'■('«'-S*C-S*«(^-S).
*-''.(l?-^')**('^'-i?)*»'(¥-5?>
20*. Bilinear covariants. Let 6^ be a linear differential form:
Gd = Xidxi + Xtdxt + • • • + JT.dx,,
where Xi, Xt, • • •, X^ are functions of the n variables Zi, Zf, • • • , z.. Let us
consider the expression
n n
where
a* __ ^-^< _ ^^k
dXk dXi
and where there are two systems of differentials, d and 8. If we make any
transformation
z. = 0.(yi, ya, •••, 2/n), (t = l, 2, ..-, n),
the expression 6^ goes over into an expression of the same form
Qd=Tidyi+--^Yndyn,
whuru I'l, I'a, •• • , F„ are functions of yi, yj, • • • , y„. Let us also aei
, 5 r, a Ft
<'-«* = -I ; —
dyt dvi
and
t it
Show that 77 = 77', identically, provided that we replace dx, and tojt, respec-
tively, by the expressions
The expression £7 is called a hUinear covaHard of 9^.
21*. Beltrami's differential parameters. If in a given expression of the form
^dx2 + 2Fdxdy ■\- Gdy^,
where E, F, G are functions of the variables x and y, we make a transformation
'•^ =/(«i f), y = <f>{Uy t), we obtain an expression of the same form:
Ex du^ + 2 Fi du dv 4- (?i di»«.
88 FUNXTIONAL RELATIONS [II, Exa
J^u Fit Oi %Te tunctiouB of u and v. "Let d{x, y) be any function of the
imriablM z and y, and ^i (u, o) the transformed function. Then we have, iden-
tieaUy,
\dx/ dx tiy \dvf _ \du / du dv \dv / ^
EG- F^ E^G^-Fl
d0 „ de\ /„ de „dd
Vmo^F* ^
[ ax £^1-4- 1 , g I gy dx \
\ ^EO - F^ I ^EG-F^ ^\ ^EG - F^ I
I du dv ] . 1 d I dv du
\ ^E, 6?, - F? /
22. Schwanijui. Setting y = (ax + 6) / (ex + 6), where x is a function of t and
a, 6, e, d are arbitrary constants, show that the relation
7f 2
2 Vx7 2/' 2 Uv
it Identically satisfied, where x', x'\ xf'\ y\ y", y"' denote the derivatives with
ratpect to the variable t.
23*. Let u and t» be any two functions of the two independent variables x and y,
and let us set
a"u + 6"» + c" ' ~ a''M + 6''t) + c"*
where a, 6, c, • • • , c" are constanta. Prove the formulae :
^g»_^aM <f^V dV d^V dU
dx^ dx ax2 dx 'd^ 'dx ~ ax^" 'dx
(Wi «) (Cr, V)
^u dv
dx* , ... .„
(M,tj)
go _ g« au /ap d^u du d^v\
dy dx'^ dy \dx dxdy 'dx dx dy)
(M,tj)
^i^_^^_U /dV ^V_ _ dU d*V \
_ ^ dy dx^ dy \dx dx dy dx dx dy)
the analogoue formulae obtained by interchanging x and y, where
dx dy dy dx' ^ ' ' dx dy dx dy'
[OouBSAT and PainlkvI:, Comptes rendus, 1887.]
CHAPTER III
TAYLOR'S SERIES ELEMENTARY APPLICATIONS
MAXIMA AND MINUdA
I. TAYLOR'S SERIES WITH A REMAINDER
TAYLOR'S SERIES
44. Taylor's series with a remainder. In elementary texts on the
Calculus it is shown that, if J\x) is an integral polynomial of
degree 7i, the following formula holds for all values of a and h:
(1) /(« + A) =/(«) + J /'(a) + ^ r{a) + . . . + ^-^/<.>(a).
This development stops of itself, since all the derivatives past the
(n -h l)th vanish. If we try to apply this formula to a function
f{x) which is not a polynomial, the second member contains an
infinite number of terms. In order to find the proper value to
assign to this development, we will first try to find an expression
for the difference
/(« + h) -f(a) - J f{a) - ^ f'ia) _^/(.)(„),
with the hypotheses that the function /(«), together with its first n
derivatives /'(x), /"(«)) • • • , /^"X*)? ^^ continuous when x lies in the
interval (a, a 4- h), and that f^*\x) itself possesses a derivative
y(« + i)^^^ in the same interval. The numbers a and a -{• h being
given, let us set
(2)
where p is any positive integer, and where P is a number which is
defined by this equation itself. L<»t us then consider the auxiliary
function
89
90 TAYLOR'S SERIES [UI,§44
^(x) =/(a + A) ^/(x) - ^±^/'(.) - i^^3^^
1.2 -n ^ ^ \ 1.2 ..n.j9 ^'
It is dear from equation (2), which defines the number P, that
^(a) = 0, <^(a + A) = 0;
and it results from the hypotheses regarding f(x) that the func-
tion ^(x) possesses a derivative. throughout the interval (a, a + h).
Hence, by RoUe's theorem, the equation <^'(a;) = must have a root
a '\- $h which lies in that interval, where d is a positive number
which lies between zero and unity. The value of <^'(x), after some
easy reductions, turns out to be
♦'("'> = ^°t.2~«''"' [^ - (« + A - xy-'^-^r-^'^x)-].
The first factor (a -\- h — xy ~ ^ cannot vanish for any value of x
other than a -f A. Hence we must have
p = A— " + » (1 - d)«-i'-^ »/(»+») (a + Oh), where < d< 1 ;
whence, substituting this value for P in equation (2), we find
(3) /(a + *)=/(a) + ^/'(a)+ j^/"(a) + . . . + _^/(»)(„)+fl,,
where
^-= 1^2...;.^ /"--'(a + OA).
We shall call this formula Taylor's series with a remainder, and
the last term or R^ the remainder. This remainder depends upon the
poeitive integer />, which we have left undetermined. In practice,
about the only values which are ever given to p are p = n + 1 and
/» ■» L Setting p = n 4- 1, we find the following expression for the
remainder, which is due to Lagrange :
^•° l .2.^(1 + 1/ '""^('' + ^">'
■ettixig/>a.l, we find
Ill, § 44] TAYLOR'S SERIKS WITH A REMAINDER 91
an expression for the remainder which is due to Cauchy. It is
clear, moreover, that the number will not be the same, in general,
in these two special formulae. If we assume further that /<""*■ *^(x)
is continuous when x = a, the remainder may be written in the form
where c approaches zero with h.
Let us consider, for definiteness, Lagrange's form. If, in the gen-
eral formula (3), n be taken equal to 2, 3, 4, • • • , successively, we
get a succession of distinct formula which give closer and closer
approximations for /(a -|- h) for small values of A. Thus for n = 1
we tind
/(« + A) =/(a) + j/'(a) + j^/" (a + tfA) ,
which shows that the difference
/(„ + A) _/(„) - ^/'(a)
is an infinitesiinal of at least the second order with respect to A,
provided that /" is finite near x = a. Likewise, the difference
is an infinitesimal of the third order ; and, in general, the expression
/(« + A) -/(«) - j/'(«) ^/*"*(«)
is an infinitesimal of order n -\- 1. But, in order to have an exact
idea of the approximation obtained by neglecting /?, we need to
know an upper limit of this remainder. Let us denote by M* an
upper limit of the absolute value of /^*"*'*>(x) in the neighborhood
of X = a, say in the interval (a — 17, a H- 1;). Then we evidently have
p |< IUlL M
provided that | A | < 17.
• That is, M > !/(•• + i){x) | when | x - a | < i». The expression *' the upper limit,"
defined in § tiS, must be carefully distinguished from the expression " an upper limit,"
which is used here to denote a number greater than or equal to the absolute Talue of
the function at any point in a certain interval. In this paragraph and in the next
/<■ + »)(») is supposed to have an upper limit near x = a. — Tbaks.
92 TAYLOR'S SERIES [III, §45
45. ApplicAtioii to curves. This result may be interpreted geomet-
rically. Suppose that we wished to study a curve C, whose equa-
tion is y =/(«), in the neighborhood of a point Aj whose abscissa
is a. Let us consider at the same time an auxiliary curve C, whose
equation is
K=/(a) + ^/'(«) + ^/»+- + fe^/-W.
A line x = a -{- h^ parallel to the axis of y, meets these two curves
in two points M and 3/', which are near A. The difference of their
ordinates, by the general formula, is equal to
This difference is an infinitesimal of order not less than w + 1 ; and
consequently, restricting ourselves to a small interval (a — rj, a -\- ?;),
the curve C sensibly coincides with the curve C". By taking larger
and larger values of n we may obtain in this way curves which
differ less and less from the given curve C; and this gives us a
more and more exact idea of the appearance of the curve near the
point A.
Let us first set n = 1. Then the curve C is the tangent to the
curve C at the point A :
r=/(a) + (x-a)/'(a);
and the difference between the ordinates of the points M and M'
of the curve and its tangent, respectively, which have the same
abtdisa a + A, is
Ut us suppose that /"(a) =^ 0, which is the case in general. The
pMceding formula may be written in the form
where c approaches zero with h. Since /"(a) =?t 0, a positive num-
ber iy can be found such that |c| < |/"(a) |, when h lies between - rj
and -f. If. For such values of h the quantity /"(a) + « will have
the aame sign as /"(a), and hence y - r will also have the same
iigli ai/"(a). Iff "(a) is positive, the ordinate y of the curve is
111,546] TAYLOR'S SERIFIS WITH A REMAINDER 98
greater than the ordinate Y of the tangent, whatever the sign of h ;
and the curve (' lies wholly above the tangent, near the point A,
On the other hand, if /"(a) is negative, y is less than K, and the
curve lies entirely below the tangent, near the point of tangenoy.
If /"(a) = 0, let /^'''(a) be the first succeeding derivative which
does not vanish for x =i a. Then we have, as before, if f^'\x) is
continuous when x = a,
y- I' =177^ [/""(«) + «];
and it can be shown, as above, that in a sufficiently small interval
(a — i;, a -I- »;) the difference y — Y has the same sign as the product
hpf(p^(^a). When j^ is even, this difference does not change sign
with hf and the curve lies entirely on the same side of the tangent,
near the point of tangency. But if p be odd, the difference y — Y
changes sign with A, and the curve C crosses its tangent at the
point of tangency. In the latter case the point A is called a point
of inflection ; it occurs, for example, if f"\a) ^ 0.
Let us now take n = 2. The curve C is in this case a parabola :
r=f(a) + (X - o)/'(a) + ^^fj^rW,
whose axis is parallel to the axis of y; and the difference of the
ordinates is
y- 1- =1:1:3 [/"'(«) + «]•
If /'"(a) does not vanish, y — Y has the same sign as h*f"'(a) for
sufficiently small values of h, and the curve C crosses the parabola
C at the point A. This parabola is called the oscuUUory parabola
to the curve C ; for, of the parabolas of the family
Y = 7nx^ -\- nx -^ Pf
this one comes nearest to coincidence with the curve C near the
point .4 (see § 213).
46. General method of development. The formula (3) affords a
method for the development of the infinitesimal f(a -\- h) —/(a)
according to ascending powers of h. But, still more generally, let
X be a principal infinitesimal, which, to avoid any ambiguity, we
94 TAYLOR'S SERIES [III, §46
will suppose positive ; and let y be another infinitesimal of the
form
(4) y = ^ix^ + A^T** 4- . . • 4- a^-'C^p + «),
where n„ «,,.••, n, are ascending positive numbers, not necessarily
integers, .4i, ^4,, •••, A^ are constants different from zero, andc is
another infinitesimal. The numbers Wi, ^i, n^, A^, •■■ may be cal-
culated successively by the following process. First of all, it is
clear that n, is equal to the order of the infinitesimal y with
respect to x, and that Ai is equal to the limit of the ratio y/x^i when
X approaches zero. Next we have
y - AiX^i = ui = AiX^'t H h (^p + €)»"',
which shows that w, is equal to the order of the infinitesimal Wi,
and At to the limit of the ratio Wi/cc"*. A continuation of this
process gives the succeeding terms. It is then clear that an infini-
tesimal y does not admit of two essentially different developments of
the form (4). If the developments have the same number of terms,
they coincide; while if one of them has p terms and the other
p •\- q terms, the terms of the first occur also in the second. This
method applies, in particular, to the development off (a -\- h) —/(a)
according to powers of h ; and it is not necessary to have obtained
the general expression for the successive derivatives of the func-
tion f(x) in advance. On the contrary, this method furnishes
us a practical means of calculating the values of the derivatives
A«). /».••••
Examples. Let us consider the equation
(5) ^i^fV) = Axr •{. By + xy^(x, y) -^ Cx-^' -{-•••+ Dy^ + "■ = 0,
where ♦ («, y) is an integral polynomial in x and y, and where the
termi not written down consist of two polynomials P(x) and Q(y),
which are divisible, respectively, by a;" + ' and y\ The coefficients A
Aod B are each supposed to be different from zero. As x approaches
wro there is one and only one root of the equation (5) which ap-
proaohea rero (§ 20). In order to apply Taylor's series with a
remainder to this root, we should have to know the successive deriv-
ativee, which could be calculated by means of the general rules.
But we may proceed more directly by employing the preceding
method. For this purpose we first observe that the principal part
Ill, §4<J] TAYLOR'S SERIES WITH A REMAINDER 95
of tlie intinitesimal root is equal to — (^4 /B)x". For if in the equa-
tion (5) we make the substitution
y
H*^)-
and then divide by j;", we obtain an equation of the same form
(6) {
^\(«» y\) = ^lX^ -f- %i -I- xy^ *i {x, y,)
-I- CiX". + '-h- •+ /^//?4-... = 0,
which has only one term in yi, namely Hy^' As a* approaches zero
the equation (6) possesses an infinitesimal root in y,, and conse-
quently the infinitesimal root of the equation (5) has the principal
part — (^//?)x'', as stated above. Likewise, the principal part of
yx is — (^li/i^)x"t; and we may set
y=-|x- + ^-^ + y,jx- + -i,
where y, is another infinitesimal whose principal part may be found
by making the substitution
y. = a-.(-|i + y.j
in the equation (6).
Continuing in this way, we may obtain for this root y an expres-
sion of the form
y = arr" -h a,x" + "t + a,x"+*t + *» ^ h («p + e)a;" + "i-»-- "^V,
which we may carry out as far as we wish. All the numbers n,
n,, n,, •••, n^ are indeed positive integers, as they should be, since
we are working under conditions where the general formula (3) is
applicable. In fact the development thus obtained is precisely the
same as that which we should find by applying Taylor's series with
a remainder, where a = and h — x.
I^et us consider a second example where the exponents are not
necessarily positive integers. I^et us set
^a;*4- 5x^4- Car>-f •••
y =
1 -f J5iX^»-|- Cix>' H
96 TAYLOR'S SERIES [III, §46
where a, ft 7, ••• and ft, yi, • • • are two ascending series of positive
nomben, and the coefficient A is not zero. It is clear that the prin-
dpal part of y is Ax^j and that we have
which is an expression of the same form as the original, and whose
principal part is simply the term of least degree in the numerator.
It is evident that we might go on to find by the same process as
many terms of the development as we wished.
Let/(x) be a function which possesses n + 1 successive derivatives. Then
replacing a by z in the formula (3), we find
/(x + A) =/(x) + ^r{x) + r^r(x) + ■ • • + 7-^ [/<">(x) + e] ,
wbere c approaches zero with h. Let us suppose, on the other hand, that we
had obtained by any process whatever another expression of the same form for
/{x + h):
/(X + h) =/(x) + h<f>i{x) + h^</>2{x) + • • • + /i" IMX) + 61.
Tbete two developments must coincide term by term, and hence the coefficients
^» ^. • • • » 0)» are equal, save for certain numerical factors, to the successive
derivatives of f(x) :
^i(x)=/'(x), 0,(x)=-^, ..., 0„(x)- -^^"^^"^^
1.2 ' ^"'' 1.2...n
This remark is sometimes useful in the calculation of the derivatives of certain
functions. Suppose, for instance, that we wished to calculate the nth derivative
of a function of a function :
y = /(u) , where u = (x) .
Neglecting the terms of order higher than n with respect to h, we have
k = ^(x + A) - 0(x) = J 0'(x) + -^ 0"(x) -f- . • + , j^" 0(«)(x);
1 1.2 1.2- n
and Ukewiee neglecting terms of order higher than n with respect to *,
1 I . Z 1 . iS • • • H
If in the right-hand side * be replaced by the expression
and the reeultlng expresalon arranged according to ascending powers of A, it is
erident that the t«rma omitted will not affect the terms in ^, ^a, . . . , A". The
III. f 47] TAYLOR'S SERIES WITH A KL^lAlM^tU 97
coefficient of A", for instance, will be equal to the nth deriymtite of /[^(x)]
diTided by 1 . 2 • • • n ; and hence we may write
where Ai denotes the coefficient of A" in the deyelopment of
For greater detail concerning this method, the reader is referred to Hermite's
Coura d' Analyse (p. 50).
47. Indeterminate forms.* Let f(x) and i>(x) be two functions
which vanish for the same value of the variable x = a. Let us try
to find the limit approached by the ratio
/(a + h)
as h approaches zero. This is merely a special case of the problem
of finding the limit approached by the ratio of two infinitesimals
The limit in question may be determined immediately if the prin-
cipal part of each of the infinitesimals is known, which is the case
whenever the formula (3) is applicable to each of the functions
f(x) and </> (x) in the neighborhood of the point a. Let us suppose
that the first derivative of f(x) which does not vanish for x = a is
that of order p, f^*'^{a) ; and that likewise the first derivative of
^ {x) which does not vanish for x = a is that of order y, ^^«^(a).
Applying the formula (3) to each of the functions /(x) and ^(x)
and dividing, we find
/(^ + A) ^ 1.2..-y/n«)-fc
where e and «' are two infinitesimals. It is clear from this result
that the given ratio increases indefinitely when h approaches zero, if
q is greater than i> ; and that it approaches zero if q is less than p,
U q =Pf however, the given ratio approaches f^''\<t)/4>^^^(o) as its
limit, and this limit is different from zero.
Indeterminate forms of this sort are sometimes encountered in finding the
tangent to a curve. Let
z =/(<), y = 0(e), t = ^(0
• See also $ 7.
98 TAYLOR'S SERIES [m,§48
be the equationa of a curve C in terms of a parameter t The equations of the
tan^eiit to this curve at a point M, which corresponds to a value U of the param-
«|«r, are, M we saw in § 5,
Z-f{to) ^ F-0(<o) ^ Zj-mKM.
/'(to) 0'(^o) V''(«o)
Thmt equaUona reduce to identities if the three derivatives /(t), 0'(t), \p'{t) aU
TaDlsh for t = t©. In order to avoid this difficulty, let us review the reasoning
by which we found the equations of the tangent. Let M' be a point of the
curve C near to 3f, and let <o + A be the corresponding value of the parameter.
Then the equationa of the secant MM' are
Z-f{to) ^ r-0(<o) ^ Z-ypjto)
/{to + A) -/(to) 0(to + A) - 0(to) ^ {to + h)- ypito)
For the sake of generality let us suppose that all the derivatives of order less
than p(p> 1) of the functions /(<), <t> (0, ^ (0 vanish for t = to, but that at least
one of the derivatives of order p, say /<p) {to), is not zero. Dividing each of the
denominators in the preceding equations by hp and applying the general for-
mula (8), we may then write these equations in the form
X-f{to) ^ T-4>{to) ^ Z-rp{to)
f^P^ (to) + e 0('» (to) + e' ^(1') (to) + c" '
where e, «', «" are three infinitesimals. If we now^ let h approach zero, these
equations become in the limit
X-f{to) ^ r-0(to) ^ Z-V^(to)
/(p)(to) 0(i')(to) ^(P)(to) '
In which form all indetermination has disappeared.
The points of a curve C where this happens are, in general, singular points
where the curve has some peculiarity of form. Thus the plane curve whose
equations are
X = t2, y = «8
through the origin, and dx/dt = dy /dt = a,t that point. The tangent
is the axis of z, and the origin is a cusp of the first kind.
48. Taylor's series. If the sequence of derivatives of the function
/(x) is unlimited in the interval (a, a + h), the number n in the
formula (3) may be taken as large as we please. If the remainder
R^ approa/ihes tero when n increases indefinitely, we are led to write
down the following formula :
(7)/(« + A) = /(a) + J/'(a)+j^r(a) + ... + j-|--/(-)(„) + ...,
which expresses that the series
m,$18] TAYLORS SERIES WITH A ROIAINDER 99
is convergent, and that its " sum " ♦ is the quantity /(a 4- A). This
formula (7) is Taylor's aeries, properly speaking. But it is not justi-
fiable unless we can show that the remainder R^ approaches zero when
n is infinite, whereas the general formula (3) assumes only the exist-
ence of the first n -f 1 derivatives. Keplacing a by as, the equation
(7) may be written in the form
/(x + A) = /(x) +*/'(*)+.. + -j-^ /<">(x) +... .
Or, again, replacing A by x and setting a = 0, we find the formula
(8) /(x) =/(0) + I /'(O) + . . . + j-^ /'•>(0) + . . ..
This latter form is often called MaclaurirCs series; but it should
be noticed that all these different forms are essentially equivalent.
The equation (8) gives the development of a function of x accord-
ing to powers of x ; the formula (7) gives the development of a func-
tion of h according to powers of A : a simple change of notation is
all that is necessary in order to pass from one to the other of these
forms.
It is only in rather specialized cases that we are able to show
that the remainder R^ approaches zero when n increases indefinitely.
If, for instance, the absolute value of any derivative whatever is less
than a fixed number M when x lies between a and a -f A, it follows,
from Lagrange's form for the remainder, that
1 A I " + >
l^^-l<-^ ^.2...(n + l) '
an inequality whose right-hand member is the general term of a
convergent series.f Such is the case, for instance, for the functions
e'j sin Xy cos x. All the derivatives of e' are themselves equal to
e*, and have, therefore, the same maximum in the interval con-
sidered. In the case of sin x and cos x the absolute values never
exceed unity. Hence t/ie formula (7) is applicable to these three
functions for all values of a and A. Let us restrict ourselves to
the form (8) and apply it first to the function f(x) = e». We find
/(0) = 1, /'(0) = 1, •.., /<">(0) = 1, ...;
* That is to say, the limit of the sura of the first n terms as n becomes Infinite.
For a definition of the meaning of the technical phrase '* the stun of a terits" see
§ 1.77. — Trans.
f The order of choios is a, h, M, n, not a, h, n, M. This is essential to the oon*
Terf^enco of the series in question. — Trans.
100
TAYLOR'S SERIES [in,§49
and Qonaequently we have the formula
(9) ^=1 + 1 + 172^ ••■^r:2T:^"^**''
which applies to all values, positive or negative, of x. If a is any
positive number, we have a' = e'>'«-, and the preceding formula
becomes
(10)
^ , _^^log£ , {xloga}_^ {xXo^ar
<»^=^+~l[ "^ 1.2 ^1.2--7i
Let us now take f{x) = sin x. The successive derivatives form a
recurrent sequence of four terms cos x, — sin x, — cos a;, sin a; ; and
their values for x = form another recurrent sequence 1, 0, — 1, 0.
Hence for any positive or negative value of x we have
X iC* ^
(11) 8inx = j-j-273 + i 2.3.4.5"*"
■^(~-^)"l.2.3. •(2714-1)'^*"*
and, similarly,
(12) eo»'' = l-0 + i.2^V4 ~--- + <~^>" l.2.3---2»t + --
Let us return to the general case. The discussion of the remain-
der R^ is seldom so easy as in the preceding examples; but the
problem is somewhat simplified by the remark that if the remain-
der approaches zero the series
/(«) + */'(«) + ••■ + jtI^ /<"'(«)+•• •
neoessarily converges. In general it is better, before examining
/J„ to see whether this series converges. If for the given values of
a and h the series diverges, it is useless to carry the discussion
further ; we can say at once that R^ does not approach zero when n
increases indefinitely.
49. Development of log(l -f x). The function log(l -f «), together
with all its derivatives, is continuous provided that x is greater
than — 1. The successive derivatives are as follows :
lll.nyj TAYLOR'S SERIES WITH A REBiAINOEB lOl
/"(') = (ItV.'
Let us see for what values of x Maclaurin's formula (8) may be
applied to this function. Writing first the series with a remainder,
we have, under any circumstances,
log(l+x) = f-| + | + -.. + (-l)— •{+/..
The remainder R^ does not approach zero unless the series
converges, which it does only for the values of x between — 1 and
-f 1, including the upper limit -f 1. When x lies in this interval
the remainder may be written in the Cauchy form as follows :
"" 1.2 -n (1-hdx)--* ^ ^ (H-ftr)-+»
or
Let us consider first the case where |x|<l. The first factor x
appi caches zero with x, and the second factor (1 ~ 6)/(l -h ftc) is
less than unity, whether x be positive or negative, for the numer-
ator is always less than the denominator. The last factor remains
finite, for it is always less than 1/(1 — \x\). Hence the remainder
R^ actually approaches zero when n increases indefinitely. This
form of the remainder gives us no information as to what happens
when X = 1 ; but if we write the remainder in Lagrange's form,
1 1
^. = (-1)"
n + l(l+$Y-^'
it is evident that A', a})proa("hes zero when n increases in(lefinit«'ly.
An exanunation of the remainder for x = — 1 would be useless,
102 TAYLOR'S SERIES fill. §49
since the series diverges for that value of x. We have then, when
X lies between — 1 and -h 1, the formula
(13) log(l + «) = f-f + |'-- + (-l)"-'f + -.
This formula still holds when a; = 1, which gives the curious
relation
(U) log2 = l-i + ^-i+- -t-(-l)-i+-.
The formula (13), not holding except when x is less than or equal
to unity, cannot be used for the calculation of logarithms of whole
numbers. Let us replace a; by — x. The new formula obtained.
(13-) log(l-x)=---
X o^ ^ _ _ ^
__ ^ ^
still holds for values of x between — 1 and -|- 1 ; and, subtracting
the corresponding sides, we find the formula
When X varies from to 1 the rational fraction (1 + a;) / (1 — x)
steadily increases from 1 to + oo, and hence we may now easily cal-
culate the logarithms of all integers. A still more rapidly con-
verging series may be obtained, however, by forming the difference
of the logarithms of two consecutive integers. For this purpose
let us set
H-x N-\-l 1
J ov x =
1-x N. 2iV-l-l
Then the preceding formula becomes
log(;.+l)-log;v=2[^ + 3-^^X_ + ^-^^ + ...].
an equation whose right-hand member is a series which converges
very rapidly, especially for large values of N.
Note, l^ lu apply the general formula (3) to the function log (1 + x), setting
a = 0, A = z, n = 1, and taking Lagrange's form for the remamder. We find in
thiiway
log(l -f x) = x-
2(1 + ftc)«
111. §4yj rWlOR'S SKUIKS WITTT V REMAINDER 108
If we uow repliicu x i . ol an uiieger n, this m«y be written
^\ n/ n 2ii«
where 9. is a positive number less than nnity. Some interesting conaeqnenoes
may be deduced from this equation.
1) The liarmonic series being divergent, the sum
inoretses indefinitely with n. But tlie difiference
2,, - log n
approaches a finite limit. For, let us write this difference in the form
(>-.o«^).g-.o.0.....(l-.o.a±i)
+ •
P I
n+ 1
Now 1 /p - log (1 + 1 /p) is the general term of a convergent series, for by the
equation above
p \ vJ 1p
2p«
which shows that this term is smaller than the general term of the convergent
series 2(1 / p^). When n increases indeiinitely the expression
n \ n/
approaches zero. Hence iho difference under consideration approaches a finite
limit, which is called Euler'a coiistant. Its exact value, to twenty places of
decimals, U C = 0.67721660490163286060.
2) Consider the expression
> +_L, + . .. + _!_.
n+1 n+2 n+p
where n and p are two positive integers which are to increase indefinitely. Then
we may write
-(■*;* ■*=f,)-(-i*-;)-
l + ^+-+~^ = l0g(»+p)+/Ni^,.
1+^ + - +^ =l0gll + /N„
104
TAYLOR'S SERIES [ill, §50
where ^+, and ^ approach the same value C when n aiid p increase indefi-
nitely. Heooe we have also
= log(l+?)
+ Pn+p - Pn
Now the difference p,+p - Pn approaches zero. Hence the sum S approaches
DO limit unle« the ratio p/n approaches a limit. If this ratio does approach a
limit a, the sum 2 approaches the limit log (1 + a).
Setting p = n, for instance, we see that the sum
iTTl "^ n + 2 ' " * 2 n
approaches the limit log 2.
50. Development of (1 + x)". The function (1 4- x)"" is defined and
continuous, and its derivatives all exist and are continuous func-
tions of aj, when 1 + « is positive, for any value of m ; for the
derivatiyes are of the same form as the given function:
/'(x)=m(l +x)''-S
/"(ar)=m(m-l)(l+x)'"-2,
m — n — 1
/<">(a;) = m(w - 1) • • • (m - 71 4- 1) (1 + ^)
/(• + i)(x) = w (w - 1) • • • (m - w) (1 4- «)
Applying the general formula (3), we find
(l+x). = l+^x + t;^l^x^ + ...
and, in order that the remainder R^ should approach zero, it is first
of all necessary that the series whose general term is
m(m — 1) • • • (m — n + 1) ^
1.2-..n *
should converge. But the ratio of any term to the preceding is
m — n 4- 1
which approaches — x as n increases indefinitely. Hence, exclud-
ing the case where m is a positive integer, which leads to the ele-
mentary binomial theorem, the series in question cannot converge
|x|< t. Let us restrict ourselves to the case in which | x { < 1.
111,53"] rvvrnRv; sKRrK>; wttif a kkmainder 105
To show thai lim luuxaiudur iii>\ zero, let us write it in the
Cauchy form :
The first factor
m(m-l) -(m-n)
1.2...n ^
approaches zero since it is the getieral term of a convergent
series. The second factor (1 — 0) /(I -\- Ox) is less than unity; and,
finally, the last factor (1 -f Ox)"'-^ is less than a fixed limit. For,
if w - 1 > 0, we have (1 + Ar)"— * < 2"*-^ ; while if 7/t - 1< 0,
(1 -I- ^)"*"'< (1 — Ixl)"-'. Hence for every value of x between
— 1 and + 1 we have the development
(16)
(1+^)^ = 14-^x4- ^<7^^> x« + ...
I rn(m-l).-.(m-n + l) ^^ ^
1 .2---n
We shall postpone the discussion of the case where x = ±1,
In the same way we might establish the following formulae :
1.3.5 ••(2n-l) g«"^^
"*■ 2.4.6..2n 2»4-l
x*x^ x'' , , . .. ar«-^» .
arc tan x = x — — + — — — H 1-(— 1)"
3 5 7 • ^ ^ 2»4-l '
which we shall prove later by a simpler process, and which hold
for all values of x between — 1 and 4- 1.
Aside from these examples and a few others, the discussion of
the remainder presents great difficulty on account of the increas-
ing complication of the successive derivatives. It would therefore
seem from this tirst examination as if the application of Taylor's
series for the development of a function in an infinite series were of
limited usefulness. Such an impression would, however, be utterly
false ; for these developments, quite to the contrary, play a funda-
mental rSle in modern Mathematical Analysis. In order to appre-
ciate their importance it is necessar}' to take another point of
view and to stuily the properties of power series for their own
1Q6 TAYLOR'S SERIES [III, § BO
sake, irrespective cf their origin. We shall do this in several of
the following chapters.
Just now we will merely remark that the series
/(o) + f /'(O) + i^/"(0) + • •• + r^/'"'(o) + • • •
may very well be convergent without representing the function
f{x) from which it was derived. The following example is due to
Cauchy. Let f{x) = e -'^< . Then f'(x) = (2/x*)e-'^'^ ; and, in
general, the nth derivative is of the form
P -\
^(n)(^)= e ^,
X"
where P is a polynomial. All these derivatives vanish for x = 0,
for the quotient of e"^^"^ by any positive power of x approaches
zero with x* Indeed, setting x = l/z, we may write
Lri
e ^ =
e^'
and it is well known that e*'/^"* increases indefinitely with z, no
matter how large m may be. Again, let ^ (x) be a function to which
the formula (8) applies :
X"
♦(0=) = ♦(<)) + J *'(0) + • • ■ + j-g:^ *<">(0) + . . •.
Setting F(x) = <>(x) + e'^"", we find
f(0) = *(0), F'(0) = *'(0), •••, F<">(0) = ,^<"'(0), ••.,
and hence the development of F(x) by Maclaurin's series would
coincide with the preceding. The sum of the series thus obtained
represents an entirely different function from that from which the
series was obtained.
In gftieral, if two distinct functions f(x) and <^ (x), together with
all their derivatives, are equal for a? = 0, it is evident that the
*It U tacitly AMnmed that /(O) = 0, which is the only assignment which would
nad»r/(x) oontlnuous at z = 0. But it should be noticed that no further assignment
b mw mu r j for /'(a-), etc., at as = 0. For
11m /(g)~/(0) _
X
/^W=a?=0 ^'r-''' =0>
which <l«aiiM/'(jr) at as ■;= and makes /'(z) continuous at z = 0, etc. —Trans.
Ill, §61] TAYLOR'S SEHIKS WITH A KKMAINDEB 107
Maclaurin series developments for the two functions cannot both
be valid, for the coefficients of the two developments coincide.
51. Extension to functions of several variables. Let us consider, for
detiniteness, a fuiictiou <i> =/(-c, y, ar) of the three independent vari-
ables Xf y, z, and let us try to develop /(ar -f A, y -H k, z -\- 1) accord-
ing to powers of h, ky /, grouping together the terms of the same
degree. Cauchy reduced this problem to the preceding by the fol-
lowing device. Let us give x, y, «, h, k, I definite values and let
us set
<^ (t) =f(x -f Af, y -f- kt, z -h U),
where t is an auxiliary variable. The function <f>(t) depends on t
alone ; if we apply to it Taylor's series with a remainder, we find
(17)
4,(t) = *(0) + J .^'(0) + j^ *"(0) + •
.» + l
^'"XO) + , o /^_.ix ^^"-^ *X^)'
1.2...n^ ^ ^ 1.2--(n-fl)
where <^(0), <^'(^)> "» <^^"X^) *^® ^^® values of the function <f>{t)
and its derivatives, for ^ = 0; and where <f>^'*'*'^\$t) is the value of
the derivativiB of order n -f- 1 for the value Ot, where 6 lies between
zero and one. But we may consider <^ (^) as a composite function of
ty ifi(t) =/(m, V, w)y the auxiliary functions
u = X i- ht, V = y -{- kt, w = z -{- It
being linear functions of t. According to a previous remark, the
expression for the differential of order m, rf*<^, is the same as if u,
V, w were the independent variables. Hence we have the symbolic
equation
\cu ov cw J \cu cv cw I
which may be written, after dividing by rff", in the form
For f = 0, u, r, w reduce, respectively, to x, y, «, and the above
equation in the same symbolism becomes
*'-<». -(g-'^- If')"'
IQg TAYLOR'S SERIES [III, §52
Similarly,
where x, y, « are to be replaced, after the expression is developed, by
X + Bht, y + 6kt, z + Bit,
lespectiTely. If we now set ^ = 1 in (17), it becomes
The remainder R^ may be written in the form
" 1.2
where a?, y, « are to be replaced hy x -\- Oh^ y ■\- Ok, z + 61 after the
expression is expanded.*
This formula (18) is exactly analogous to the general formula
(3). If for a given set of values of x, y, z, h, k, I the remainder R^
approaches zero when n increases indefinitely, we have a develop-
ment of fix + A, y 4- A;, « H- Z) in a series each of whose terms is a
homogeneous polynomial in A, k, I. But it is very difficult, in gen-
eral, to see from the expression for R^ whether or not this remainder
approaches zero.
52. From the formula (18) it is easy to draw certain conclusions
analogous to those obtained from the general formula (3) in the
case of a single independent variable. For instance, let z =f(x, y)
be the equation of a surface S. If the function /(x^ y), together
with all its partial derivatives up to a certain order n, is continuous
in the neighborhood of a point (xq, yo), the formula (18) gives
A*+»,y.+*)-A..,,.) + (ig + *g)
Reatricting oumelyes, in the second member, to the first two terms,
then to the first three, etc., we obtain the equation of a plane, then
• It IsMMUMd here that all the derlvatlvea used exist and are continuous. — Trans.
in. §52] TAYLOR'S SERIES WITH A REMAINDER 109
that of a paralx)loid, etc., which differ very little from the given sur-
face near the point (xq, yo). The plane in question is precisely the
tangent plane ; and the paraboloid is that one of the family
« = ilx* + 2 Bxy -\-Cy* + 2Dx + 2Ei/-\-F
which most nearly coincides with the given surface 5.
The formula (18) is also used to determine the limiting value of
a function which is given in indeterminate form. Let /(x, y) and
<^ (x, y) be two functions which both vanish for x = a, y = by but
which, together with their partial derivatives up to a certain order,
are continuous near the point (a, b). Let us try to find the limit
approached by the ratio
f(^. y)
when X and y approach a and 6, respectively. Supposing, first, that
the four first derivatives df/da, df/dbj dif»/day c<^/db do not all
vanish simultaneously, we may write
•(K-)-(l'-')
f(a -{.h,b-\-k) _
it,(a -{- h, b -\- k) /d<f> \ /cif>
(5*-.)"(S-0
where e, e', c,, «[ approach zero with h and k. When the point
(ar, y) approaches (a, b), h and k approach zero ; and we will sup-
pose that the ratio k/h approaches a certain limit a, i.e. that the
point {Xy y) describes a curve which has a tangent at the point (a, b).
Dividing each of the terms of the preceding ratio by A, it appears
that the fraction /(x, y)/<f>(Xf y) approaches the limit
da db
da db
This limit depends, in general, upon <r, i.e. upon the manner in
which X and y approach their limits a and i, respectively. In order
that this limit should be independent of a it is necessary that the
relation
da db db da
should hold ; and such is not the case in generaL
IIQ TAYLOR'S SERIES [III, §63
If the four first derivatives a//aa, dfj^h, d^jda, d4>/dh vanish
simultaneously, we should take the terms of the second order in the
formula (18) and write
where c, c', c", €„€,', c'/ are infinitesimals. Then, if a be given the
same meaning as above, the limit of the left-hand side is seen to be
which depends, in general, upon a.
n. SINGULAR POINTS MAXIMA AND MINIMA
53. Singular points. Let (xqj yo) be the coordinates of a point Mq
of a curve C whose equation is F(x, y) = 0. If the two first par-
tial derivatives dF/dx, dF /dy do not vanish simultaneously at this
point, we have seen (§ 22) that a single branch of the curve C passes
through the point, and that the equation of the tangent at that
point is
where the symbol d^-^'^F/dxl dyl denotes the value of the derivative
a^ ♦«F/3a?»' dj/f for a; = Xo, y = yo- If dF/dxo and dF/dy^ both van-
ish, the point (^o. y©) is, in general, a singular point* Let us suppose
that the three second derivatives do not all vanish simultaneously
for x^X9,y = yo» aiid that these derivatives, together with the third
deriTatiTes, are continuous near that point. Then the equation of
the ounre may be written in the form.
* That to, tlM ftppeanncA of the curve is, in general, peculiar at that point. For an
aaalytJe dafloitloD of a ringular point, see $ 192. — Tkans.
Ill, §wj SINGULAR POINTS MAXIMA AND MINIMA 111
ifd^F d*F d^F "I
1 VdP dp T"
+ r2li\ji ^' -''^ + Vy^!'- ^">J;;:jji;j;'
where x and 3/ are to be replaced in the third derivatives by
Xo-\-$(x — Xo) and yo-h 0(i/ — yo)y respectively. We may assume
that the derivative d^F/dyl does not vanish; for, at any rate, we
could always bring this about by a change of axes. Then, setting
y — yo = t(^ — ^o) and dividing by (x — Xoy, the equation (19)
becomes
d^F d*F d^F
(20) a;^ + 2.a^^ + ''ai^+(— »)^(--'^'') = «'
where P{x — XQjt) is a function which remains finite when x
approaches Xq. Now let ti and t^ be the two roots of the equation
axj axo dy^ dy^
If these roots are real and unequal, i.e. if
/ d^F y d^Fd^F
\dxQ dyj cxl dyi '
the equation (20) may be written in the form
^(^ - ty){t - U) ^{X-X,)P = 0.
For X = Xq the above quadratic has two distinct roots < = ^1, t = t^.
As X approaches 2*0 that equation has two roots which approach ti
and fiy respectively. The proof of this is merely a repetition of
the argument for the existence of implicit functions. Let us set
t = ti -\- u, for example, and write down the equation connecting x
and u :
u(h _ ^, 4. u) -f (x - Xo) (2(x, tt) = 0,
where Q(x, u) remains finite, while x approaches x© and u approaches
zero. Let us suppose, for definiteness, that t^ — tt>0; and let M
denote an upper limit of the absolute value of Q(x, t/), and m a
lower limit of fi — ^j -f u, when x lies between x© — A and x© -f ^
112 TAYLOR'S SERIES [III, §53
and 11 between — h and 4- A, where A is a positive number less than
tx — tf. Now let c be a positive number less than h, and rj another
positive Dumber which satisfies the two inequalities
If :c be given such a value that |a; — xd is less than rj, the left-hand
side of the above equation will have different signs if — c and then
4- c be substituted for u. Hence that equation has a root which
approaches zero as x approaches Xq, and the equation (19) has a
root of the form
y = yo 4- (a; - Xo) (tt -f- or),
where a approaches zero with x — Xq. It follows that there is one
branch of the curve C which is tangent to the straight line
y — yo = h(x- Xo)
at the point (x©, y©)-
In like manner it is easy to see that another branch of the
curve passes through this same point tangent to the straight line
y — t/o ~ ^t(^ — ^o)' The point Mq is called a double point; and
the equation of the system of tangents at this point may be found
by setting the terms of the second degree in (x — Xq), Q/ — y^) in
(19) equal to zero.
If
\dx,dyj dxl dif.
<o,
the point (ajo, y©) is called an isolated double point. Inside a suffi-
ciently small circle about the point Mq as center the first member
F(ar, y) of the equation (19) does not vanish except at the point Mq
itself. For, let us take
35 = a^o + p cos <^, y = ?/o + /» sin <^
as the cottrdinates of a point near M^. Then we find
where L remains finite when p approaches zero. Let // be an upper
limit of the absolute value of L when p is less than a certain posi-
tive number r. For all values of <^ between and 27r the expression
d^F , ^ d^F d^F
in,§53] SINGULAR POINTS MAXIMA AND MINIMA 118
has the same sign, since its rcx)t8 are imaginary. Let m be a lower
limit of its absolute value. Then it is clear that the coefficient
of p* cannot vanish for any point inside a circle of radius p<m/H.
Hence the equation F(x, y) = has no root other than /> = 0, i.e.
X = a;©, y = 2/o> inside this circle.
In case we have
F
/_?!£. V= ^ ?H
the two tangents at the double point coincide, and there are, in gen-
eral, two branches of the given curve tangent to the same line, thus
forming a cusp. The exhaustive study of this case is somewhat
intricate and will be left until later. Just now we will merely
remark that the variety of cases which may arise is much greater
than in the two cases which we have just discussed, as will be seen
from the following examples.
The curve y^ = x* has a cusp of the first kind at the origin, both
branches of the curve being tangent to the axis of x and lying on
different sides of this tangent, to the right of the y axis. The
curve y^ — 2 x^y -\- X* — x^ =^ has a cusp of the second kindj both
branches of the curve being tangent to the axis of x and lying on
the same side of this tangent; for the equation may be written
y = x^ dt x*f
and the two values of y have the same sign when x is very small,
but are not real unless x is positive. The curve
x* -{- x^y^ - 6 x^y -\- y^ =
has two branches tangent to the x axis at the origin, which do not
possess any other peculiarity ; for, solving for y, the equation becomes
3x*±x^y/S-x*
2^ = iTV^ '
and neither of the two branches corresponding to the two signs
before the radical has any singularity whatever at the origin.
It may also happen that a curve is composed of two coincident
branches. Such is the case for the curve represented by the
equation
F{x,y) = y^-2x*y + x*==0.
When the point (x, y) passes across the curve the first member F(x, y)
vanishes without changing sign.
114 TAYLOR'S SERIES [III, §54
Finally, the point (xo, yo) may be an isolated double point. Such
is the case for the curve 1/ -\- x* + y* = 0, on which the origin is an
isolated double point.
54. In like manner a point Mq of a surface 5, whose equation is
F(Xy y, «) = 0, is, in general, a singular point of that surface if the
three first partial derivatives vanish for the coordinates Xq, ijqj Zq of
that point:
|£ = o, 1^ = 0, 1^ = 0.
dxQ dy^ czq
The equation of the tangent plane found above (§ 22) then reduces
to an identity ; and if the six second partial derivatives do not all
vanish at the same point, the locus of the tangents to all curves on
the surface S through the point Mq is, in general, a cone of the
second order. For, let
be the equations of a curve C on the surface S. Then the three
functions f(t)f <f>(t), \l/(t) satisfy the equation F(x, y, z) = 0, and
the first and second differentials satisfy the two relations
dF dF dF
(dp dF dF \"' dF dF dF
For the point x = Xq, y = y^, z = Zq the first of these equations
reduces to an identity, and the second becomes
3*F d^F d^F
^ ^yi ^4
^ d*F d*F d^F
"*■ ^ ?r^ dxdy + 2 r— r- dyd:i^2 ^-^ dx dz = 0.
The equation of the locus of the tangents is given by eliminating
dXf dy, dx between the latter equation and the equation of a tangent
line
dx dy dz ^
which leads to the equation of a cone T of the second degree :
in, 5 54] SINGULAR POINTS MAXIMA AND MINIMA 115
(21)
On the other hand, applying Taylor's series with a remainder
and carrying the development to terms of the third order, the equa-
tion of the surface becomes
(22)
1 V»F 8F dF T"
where ar, y, z in the terms of the third order are to be replaced by
Xq-\- ${x — Xo)f yo + 6{ij — i/o), Zo + 6(z — Zq), respectively. The
equation of the cone T may be obtained by setting the terms of
the second degree in x — a*©, y — yoj z — ZqIu the equation (22) equal
to zero.
Let us then, first, suppose that the equation (21) represents a real
non-degenerate cone. Let the surface .S and the cone T be cut by a
plane P which passes through two distinct generators (i and G' of
the cone. In order to find the equation of the section of the sur-
face S by this plane, let us imagine a transformation of coordinates
carried out which changes the plane /* into a plane parallel to the
xy plane. It is then sufficient to substitute » = Zom the equation (22).
It is evident that for this curve the point Mq is a double point with
real tangents ; from what we have just seen, this section is composed
of two branches tangent, respectively, to the two generators G, 6".
The surface S near the point 3/o therefore resembles the two nappes
of a cone of the second degree near its vertex. Hence the point i/©
is called a conical point.
When the equation (21) represents an imaginary non-degenerate
cone, the point ^fo is an isolated singular point of the surface ^'.
Inside a sufficiently small sphere about such a point there exists no
set of solutions of the equation F(x, y, «) = other than x = Xq,
y = yo> « = -0- For? let M be a point in space near 3/©, p the
116 TAYLOR'S SERIES [III, §55
distance AfA/o, and a, /8, y the direction cosines of the line MqM.
Then if we substitute
the function F(Xf y, z) becomes
where L remains finite when p approaches zero. Since the equation
(21) represents an imaginary cone, the expression
cannot vanish when the point (a, ft y) describes the sphere
a'^ + iS" + y' = 1.
Let m be a lower limit of the absolute value of this polynomial,
and let i/ be an upper limit of the absolute value of L near the
point Mq. If a sphere of radius m/H he drawn about Mq as center,
it is evident that the coefficient of p^ in the expression for F(x, y, z)
oannot vanish inside this sphere. Hence the equation
F(x, y, «) =
has no root except p = 0.
When the equation (21) represents two distinct real planes, two
nappes of the given surface pass through the point Mq, each of
which is tangent to one of the planes. Certain surfaces have a
line of double points, at each of which the tangent cone degenerates
into two planes. This line is a double curve on the surface along
which two distinct nappes cross each other. For example, the circle
whose equations are « = 0, x* + y* = 1 is a double line on the surface
whoee equation is
«* 4- 2z^(x* + y2) _ (a;2 + y2 _ ly ^ q.
When the equation (21) represents a system of two conjugate
im ag inar y planes or a double real plane, a special investigation is
neoetsary in each particular case to determine the form of the sur-
face near the point 3/©. The above discussion will be renewed in
the paragraphs on extrema.
66. Extrema of functions of a single variable. Let the function f(x)
be oontinuuus in the interval (a, b), and let c be a point of that
I1I,$MJ SINGULAR POINTS MAXIMA AND MINIMA 117
interval. The function f{x) is said to have an eoUremum (Le. a
maximum or a minimum) for 05 = c if a positive number iy can be
found such that the difference f{c -\- h) —f{c), which vanishes for
A = 0, has the same sign for all other values of h between — 17
and -f -q. If this difference is positive, the function f(x) has a
smaller value for x = c than for any value of x near c; it is said
to have a minimum at that point. On the contrary, if the differ-
ence /(c -f h) —f{c) is negative, the function is said to have a
maximum.
If the function f{x) possesses a derivative for x = c, that deriva-
tive must vanish. For the two quotients
nc + h)-f{c) f(c-.h)-f(c)
h -h
each of which approaches the limit /'(c) when h approaches zero, have
different signs ; hence their common limit /'(c) must be zero. Con-
versely, let c be a root of the equation /'(x) = which lies between
a and b, and let us suppose, for the sake of generality, that the
first derivative which does not vanish for x = c is that of order n,
and that this derivative is continuous when x = c. Then Taylor's
series with a remainder, if we stop with 71 terms, gives
/(<= + A) -/(«) = r:|!^/*"H<' + «A),
which may be written in the form
• /(« + A) -/(c) = j-|^ [/<•>(.) + .],
where c approaches zero with A. Let ly be a positive number such
that \f^*\c) I is greater than c when x lies between c — 1; and c -f 17.
For such values of x, /^"^c) -}- c has the same sign as f^*^(c), and
consequently /(c -f h) —f(c) has the same sign as A"/<"^(c). If
n is odd, it is clear that this difference changes sign with A, and
there is neither a maximum nor a minimum at x = c. If n is even,
/(c -}- A)— /(c) has the same sign as/^">(c), whether A be positive
or negative ; hence the function is a maximum if /^"^(c) is negative,
and a minimum if f^*\c) is positive. It follows that the necessary
and sufficient condition that the function /(x) should have a maximum
or a minimum f or aj = c is that the first derivative which does not
vanish f or x == c sliould be of even order.
118 TAYLOR'S SERIES [III, §56
Geometrically, the preceding conditions mean that the tangent to
the curve y =/(a?) at the point A whose abscissa is c must be par-
allel to the axis of ar, and moreover that the point A must not be
a point of inflection.
NUm, When the hypotheses which we have made are not satisfied
the function f{x) may have a maximum or a minimum, although
the derivative /'(x) does not vanish. If, for instance, the derivative
is infinite for a; = c, the function will have a maximum or a mini-
mum if the derivative changes sign. Thus the function y = x^ is at
a minimum for a; = 0, and the corresponding curve has a cusp at the
origin, the tangent being the y axis.
When, as in the statement of the problem, the variable x is
restricted to values which lie between two limits a and ^, it may
happen that the function has its absolute maxima and minima pre-
cisely at these limiting points, although the derivative /'(a:r) does
not vanish there. Suppose, for instance, that we wished to find
the shortest distance from a point P whose coordinates are (a, 0)
to a circle C whose equation is x^ -\- y'^ — R^ =i 0. Choosing for our
independent variable the abscissa of a point M of the circle C, we
find
(? = pF^ = (a; - a)2 + y2 ^ x2 4- y2 _ 2 «a; H- a" ,
or, making use of the equation of the circle,
d^ = R''^-a'-2ax.
The general rule would lead us to try to find the roots of the derived
equation 2 a = 0, which is absurd. But the paradox is explained if
we observe that by the very nature of the problem the variable x
must lie between — R and -f 72. If a is positive, d?^ has a minimum
iox x = R and a maximum for a; = — R.
56. Eztrema of functions of two variables. Let /(x, y) be a con-
tinuous function of x and y when the point A/, whose coordinates
are x and y, lies inside a region O bounded by a contour C. The
function f{x, y) is said to have an extremum at the point M^ (a;©, yo)
of the region O if a positive number y^ can be found such that the
diffeMnoe
^ =/(a;o -I- h, yo + k) -f(xoy yo),
which Tanishes for h = k = 0, keeps the same sign for all other sets
of values of the increments A and k which are each less than ,; in
lil,§5«ij SINGLLAK I'OINTS MAXIMA AND MINIMA 119
absolute value. Considering y for the moment as constant and
equal to yo> « becomes a function of tlie single variable x ; and, by
the above, the difference
/(xo 4- A, yo) -/(xo, yo)
cannot keep the same sign for small values of h unless the deriva-
tive df/dx vanishes at the point M^. Likewise, the derivative df/dy
must vanish at Mq ; and it is apparent tliat the only possible sets of
values of x and y which can render the function /(ar, y) an extre-
mum are to be found among the solutions of the two simultaneous
equations
ox cy
Let a; = Xq, y = yo ^ a- set of solutions of these two equations.
We shall suppose that the second partial derivatives of /(x, y) do
not all vanish simultaneously at the point 3/o whose coordinates
are (xq, y^), and that they, together with the third derivatives, are
all continuous near 3/y. Then we have, from Taylor's expansion,
^ =/(^o -\-h,y^ + k) -/(j-o, yo)
(23) \ 1.2\ dxi «x^cy„ cyi/
We can foresee that the expression
cxl ox^cy^ cy^
will, in general, dominate the whole discussion.
In order that there be an extremum at Mq it is necessary and
sufficient that the ditference A should have the same sign when the
point (xq -f a, yo + k) lies anywhere inside a sufficiently small square
drawn about the point Mq as center, except at the center, where
A = 0. Hence A must also have the same sign when the point
(xq -f- a, yo + k) lies anywhere inside a sufficiently small circle whose
center is 3fo; for such a square may always be replaced by its
inscribed circle, and conversely. Then let C be a circle of radius
/' drawn about the point Mq as center. All the points inside this
circle are given by
h = pcos<f>, k = psui<f>.
120 TAYLOR'S SERIES [III, §56
if-here ^ is to vary from to 2 tt, and p from — r to + r. We might,
indeed, restrict p to positive values, but it is better in what follows
not to introduce this restriction. Making this substitution, the
expression for A becomes
A = ^ (^ C08*«^ -f 2 J5 sin <^ cos <^ + C* sin^c^) + ^ ^
where
and where Z, is a function whose extended expression it would be
useless to write out, but which remains finite near the point (iCo, yo)«
It now becomes necessary to distinguish several cases according to
thesignof B«-^C.
Firtt case. Let B^ — AC >0. Then the equation
A cos^<l> + 2 B sin <^ cos <^ + C sin^<^ =
has two real roots in tan <f>, and the first member is the difference
of two squares. Hence we may write
^ = 2 ['^ (" cos <^ + ^ sin </>)2 - I3(a' cos <f> -\- b' sin <^)2] +^L,
where
a: > 0, ^ > 0, ab'- ba' ^ 0.
If ^ be given a value which satisfies the equation
a cos <^ + J sin <^ = 0,
A will be negative for sufficiently small values of p ; while, if <^ be
such that a'cos<^ -1- 6'sin<^ = 0, A will be positive for infinitesimal
yalues of p. Hence no number r can be found such that the differ-
ence A has the same sign for any value of <^ when p is less than r.
It follows that the function f(x, y) has neither a maximum nor a
minimum for x = Xoj y = y^.
Seeandease. LetB*- AC<0. The expression
A co8*<t> -f 2 fi cos<^ 8in<^ + C 8in*<^
oannot vanish for any value of <!>. Let m be a lower limit of its
absolute value, and, moreover, let // be an upper limit of the abso-
lute value of tlie funcrtion L in a circle of radius K about (aro, yo) aa
^V^^O,
III, $57] SiMiLLAU I'UlNTS M.VAIMA AM) M1M31.\ 121
center. Finally, let r denote a positive number less than R and less
than 3m/ II. Then inside a circle of radius r the difference A will
have the same sign a^ the coefiicient of p'j i.e. the same sign as A
or ( '. Hence the function /(x, y) has either a mail mum or a mini-
iiiuin for X = Xq, y = yo.
To recapitulate, if at the point (a;©, yo) we have
\^XodyJ dxi dyi '
there is neither a maximum nor a minimum. But if
/jyy
Y^XodyJ dxl dyi
there is either a maximum or a minimum, depending on the sign of
the two derivatives ^//^xj* ^f/^^- There is a maximum if these
derivatives are negative, a minimum if they are positive.
57. The ambiguous case. The case where B^ — AC = is not cov-
ered by the preceding discussion. The geometrical interpretation
shows why there should be difficulty in this case. Let S be the
surface represented by the e(iuation ;;; =/(a", y). If the function
f(x, y) has a maximum or a minimum at the point (x^, yo), near
which the function and its derivatives are continuous, we must have
which shows that the tangent plane to the surface S at the point
iUo, whose coordinates are (xq, yo, Zq)^ must be parallel to the xy
plane. In order that there should be a maximum or a minimum it
is also necessary that the surface 5, near the point Mq, should lie
entirely on one side of the tangent plane ; hence we are led to study
the behavior of a surface with respect to its tangent plane near the
point of tangency.
Let us suppose that the point of tangency has been moved to the
origin and that the tangent plane is the xy plane. Then the equa-
tion of the surface is of the form
(24) z = ax^-\- 2bxy + cy« + ax* -f Spx^y + 3 yxy* + 8/,
where a, b, r are constants, and where a, p, y, S are functions of x
and y which remain finite when x and y approach zero. This equa-
tion is essentially the same as equation (19), where Xo and y© have
been replaced by zeros, and h and k by x and y, respectively.
122 TAYLOR'S SERIES [in,§W
In order to see whether or not the surface S lies entirely on
one side of the xi/ plane near the origin, it is sufficient to study the
section of the surface by that plane. This section is given by the
equation
(25) ax^ -\-2bxi/ + cy^ + ax*+-- = 0j
hence it has a double point at the origin of coordinates. If b^ — ac
is negative, the origin is an isolated double point (§ 53), and the
equation (25) has no solution except x = y = 0, when the point
(x, y) lies inside a circle C of sufficiently small radius r drawn
about the origin as center. The left-hand side of the equation (25)
keeps the same sign as long as the point (a;, y) remains inside this
circle, and all the points of the surface S which project into the
interior of the circle C are on the same side of the xy plane except
the origin itself. In this case there is an extremum, and the por-
tion of the surface S near the origin resembles a portion of a sphere
or an ellipsoid.
If b^ — ac> Of the intersection of the surface S by its tangent
plane has two distinct branches Cu C^ which pass through the
origin, and the tangents to these two branches are given by the
equation
ax^ + 2 bxy -f- cy^ = 0.
Let the point (x, y) be allowed to move about in the neighborhood
of tlie origin. As it crosses either of the two branches Ci, Cg, the
left-hand side of the equation (25) vanishes and changes sign.
Hence, assigning to each region of the plane in the neighborhood
of the origin the sign of the left-hand side of the equation (25), we
find a configuration similar to Fig. 7. Among the points of the
surface which project into points inside a circle about the origin in
the xy plane there are always some which
lie below and some which lie above the
xy plane, no matter how small the circle
be taken. The general aspect of the sur-
face at this point with respect to its tan-
gent plane resembles that of an unparted
hyperboloid or an hyperbolic paraboloid.
The function /(«, y) has neither a maxi-
mum nor a minimum at the origin.
The oase where &• — a<; = is the case in which the curve of
interseotion of the surface by its tangent plane has a cusp at the
origin. We will postpone the detailed discussion of this case. If the
Fio.7
111,558] SINGULAR POINTS MAXIMA AND MINIMA 128
intersection is composed of two distinct branches through the origin,
there can be no extremiun, for the surface again cuts the tangent
plane. If the origin is an isolated double point, the function /(x, y)
has an extremum for x = y = 0. It may also happen that the inter-
section of the surface with its tangent plane is composed of two
coincident branches. For example, the surface z = y* — 2 x*y -f x*
is tangent to the plane « = all along the parabola y = x*. The
function y* — 2 x^y 4- x* is zero at every point on this parabola, but is
positive for all points near the origin which are not on the parabola.
88. In order to see which of these cases holds in a given example it is neces-
Kury to take into account the derivatives of the third and fourth orders, and some-
times derivatives of still higher order. The following discussion, which is usually
sufficient in practice, is applicable only in the most general cases. When
6* — oc = the equation of the surface may be written in the following form
by using Taylor's development to terms of the fourth order:
(26) z =/(x, y) = A (xsinw - ycosu,)^ + i>s{x, y) + -\ (»ii + v!r)»x-
z-i \ cz cy / gg
Let us suppose, for definiteness, that A is positive. In order that the surface S
should lie entirely on one side of the xy plane near the origin, it is necessary that
all the curves of intersection of the surface by planes through the z axis should
lie on the same side of the xy plane near the origin. But if the surface be cut
by the secant plane
y = ztan0,
the equation of the curve of intersection is found by making the substitution
x = p cos 0, y = psm<f>
in the equation (26), the new axes being the old z axis and the trace of the secant
plane on the xy plane. Performing this operation, we find
z = Afl^ (cos sin w — cos « sin 0)^ + Kp* + L^,
where K is independent of p. If tan « ^ tan 0, z is positive for sufficiently small
values of p ; hence all the corresponding sections lie above the xy plane near the
origin. Let us now cut the surface by the plane
y = x tan m.
If the corresponding value of X is not zero, the development of e is of the form
z = pfi{K + t)
and changes sign with p. Hence the section of the surface by this plane has a
point of inflection at the origin and crosses the xy plane. It follows that the
function /(x, y) has neither a maximum nor a minimum at the origin. Such is
the case when the section of the surface by its tangent plane has a cusp of the
first kind, for instance, for the surface
« = y« - x«.
124 TAYLOR'S SERIES [III, §68
If JT = for the latter lubstitution, we would carry the development out to
tann* of the fourth order, and we would obtain an expression of the form
where ITi ii a constant which may be readily calculated from the derivatives of
the fourth order. We shall suppose that Ki is not zero. For infinitesimal val-
ues of pt t has the same sign as iTi ; if Ki is negative, the section in question lies
beneath the zy plane near the origin, and again there is neither a maximum nor
a minimum. Such is the case, for example, for the surface z = y^ — x*, whose
intenection with the xy plane consists of the two parabolas y = ±x'^. Hence,
ntili^a K = and iri>0 at the same time, it is evidently useless to carry the
inresUgation farther, for we may conclude at once that the surface crosses its
tangent plane near the origin.
But if £* = and Ki>Q at the same time, all the sections made by planes
throng the z axis lie above the xy plane near the origin. But that does not
■how conclusively that the surface does not cross its tangent plane, as is seen
by considering the particular surface
z = {y-x'^){y-2x%
wliich cuts its tangent plane in two parabolas, one of which lies inside the other.
In order tliat the surface should not cross its tangent plane it is also necessary
that the section of the surface made by any cylinder whatever which passes
through the z axis should lie wholly above the xy plane. Let y = <f>{x) be the
equation of the trace of this cylinder upon the xy plane, where {x) vanishes for
X = 0. The function F{x) =/[x, <f>{x)] must be at a minimum for a; = 0, what-
ever be the function <f> (x). In order to simplify the calculation we will suppose
that the axes have been so chosen that the equation of the surface is of the form
where A is positive. With this system of axes we have
««o ' ^0 ' dxl ' dxodyo ' dyl '
at the origin.
The derivatives of the function F{x) are given by the formulae
Ul,$fiO] SINGULAR POINTS MAXUiA AND MINIMA 125
from which, for x = y = 0, we obtain
F'(0) = 0, F"(0) = ^[^'(0)]«.
If 0'(O) does not yanlsb, the function F(x) has a minimum^ a« is aljo apparent
from the previous discussion. But if 4>'(0) = 0, we find the formula
^(0) = 0, F"(0) = 0, F'"(0) = ^,
Hence, in order that F(x) be at a minimum, it is necessary that d^f/ta^ vanish
and that the following quadratic form in 0"(O),
be positive for all values of 4>^'{0).
It is easy to show that these conditions are not satisfied for the above function
« = y* — 3x'*y + 2x*, but that they are satisfied for the function « = y* + a:*.
It is evident, in fact, that the latter surface lies entirely above the xy plane.
We shall not attempt to carry the discussion farther, for it requires extremely
nice reasoning to render it absolutely rigorous. The reader who wishes to exam-
ine the subject in greater detail is referred to an important memoir by Ludwig
Schefler, in Vol. XXXV of the Mathematische Annalen.
59. Functions of three variables. Let u = f(x, t/, z) he 2l continuoiia
function of the three variables a*, y, z. Then, as before, this func-
tion is said to have an extremum (maximum or minimum) for a set
of values Xq, i/o, Zq if a positive number ri can be found so small
that the difference
■ ^ =/(a'o + h, 2/0 + A', ^0 4- -/(a-o, yo, «o),
which vanishes for A = A; = Z = 0, has the same sign for all other
sets of values of A, k^ /, each of which is less in absolute value
than -q. If only one of the variables a*, y, z is given an increment,
while the other two are regarded as constants, we find, as above,
that tt cannot be at an extremum unless the equations
are all satisfied, provided, of course, that these derivatives are con-
tinuous near the point (jr^, y^, «„). Let us now suppose that x^, yo» *•
are a set of solutions of these equations, and let Mq be the point
whose coordinates are j-q, %, Zq. There will be an extremum if a
sphere can be drawn about M^ so small that /(x, y, x) — f{x^y y©, «J
126 TAYLOR'S SERIES [III, §59
has the same sign for all points (x, y, z) except Mo inside the sphere.
Let the coordinates of a neighboring point be represented by the
equations
X = aro + par, y = y© + pA « = «o + py,
where a, A y satisfy the relation a« + /3^ + / = 1 ; and let us replace
a; _ Xo, y - yo, « - «o in Taylor's expansion of /(a;, y, «) by pa, p/3,
py, respectively. This gives the foUowing expression for A :
A = p*[<^(a, Ay) + p^],
where <^(a, ft y) denotes a quadratic form in a, ft y whose coeffi-
cients are the second derivatives of f(x, y, ^), and where Z is a
function which remains finite near the point M^. The quadratic
form may be expressed as the sum of the squares of three distinct
linear functions of a, ft y, say P, P', P", multiplied by certain con-
stant factors a, a', a", except in the particular case when the dis-
criminant of the form is zero. Hence we may write, in general.
<f>(cx, ft y) = aP^ + a'P'^ + a'^P"'
where a, a', a" are all different from zero. If the coefficients a, a', a"
have the same sign, the absolute value of the quadratic form </> will
remain greater than a certain lower limit when the point a, ft y
describes the sphere
a' + i8* + y' = 1,
and accordingly A has the same sign as a, a', a" when p is less than
a certain number. Hence the function f(x, y, z) has an extremum.
If the three coefficients a, a\ a" do not all have the same sign,
there will be neither a maximum nor a minimum. Suppose, for
example, that a > 0, a' < 0, and let us take values of a, ft y which
satisfy the equations P' = 0, P" = 0. These values cannot cause P
to vanish, and A will be positive for small values of p. But if, on
the other hand, values be taken for a, ft y which satisfy the equa-
tions P = 0, P" = 0, A will be negative for small values of p.
The method is the same for any number of independent variables :
the discussion of a certain quadratic form always plays the prin-
cipal rale. In the case of a function u =f(x, y, z) of only three
independent variables it may be noticed that the discussion is
equivalent to the discussion of the nature of a surface near a singu-
lar point. For consider a surface 2 whose equation is
F(«, y, «) =/(x, y, «) -/(a^o, yo, «o) = ;
UI,J60J SINGULAR POINTS MAXIMA AND MINIMA 127
this surface evidently passes through the point i/« whose co6rdi*
nates are (Xo, yo> «o)> and if the function /(x, y, z) has an extremum
there, the point ^f^, is a singular point of 2. Hence, if the cone of
tangents at Mo is imaginary, it is clear that F(jr, y, z) will keep the
same sign inside a sufficiently small sphere about J/q as center, and
/{^y !/} «) will surely have a maximum or a minimum. But if the
cone of tangents is real, or is composed of two real distinct planes,
several nappes of the surface pass through A/o, and F(x, y, z)
changes sign as the point (sc, y, z) crosses one of these nappes.
60. Distance from a point to a surface. r..et us try to find the maximum and the
ininimuin values of tlie diKUiiice from a Hzed point (a, 6, c) to a surface S whoM
equation is F(x, y, z) = 0. The square of this distance,
u = da = (z - a)2 + (y - 6)« + (2 - c)«,
is a function of two indei)endent variables only, — z and y, for example, if « be
considered as a function of x and y defined by the equation F = 0. In order
that u be at an extremum for a point (z, y, z) of the surface, we must have, for
the coordinates of that point,
2 dy cy
We find, in addition, from the equation F = 0, the relations
dx dz dx~ ^ dy dz dy~ ^
whence the preceding equations take the form
z — a y — b _ z — c
dF ~ dJF d_F '
dz dy dz
This shows that the normal to the surface iS at the point (x, y, z) passes through
tlie point (a, 6, c). Hence, omitting the singular points of the surface 5, the
points sought for are the feet of normals let fall from the point (a, 6, c) upon the
surface S. In order to see whether such a point actually corresponds to a maxi-
mum or to a minimum, let us take the point as origin and the tangent plane as
the xy plane, so that the given point shall lie upon the axis of z. Then the func-
tion to be studied has the form
u = z« + y2 + (z - c)«,
where z is a function of z and y which, together with both its first derivatives,
vanishes for x = y = 0. Denoting the second partial derivatives of z by r, s, t,
we have, at the origin,
128 TAYLOR'S SERIES [ni,§61
and it only remains to study the polynomial
A(c) = c«ja - (1 - cr)(l - ct) = c^8^ -rt) + {r-ht)c - 1.
The root* of the equation A(c) = are always real by virtue of the identity
/|. 4. t)« ^. 4 («« - rf ) = 4 «« + (r - t)^. There are now several cases which must
be distinguished according to the sign of a^ - rt.
FMt case. Let »« - rf < 0. The two roots Ci and Ca of the equation A (c) =
have the same sign, and we may write A (c) = (s^ - rt) (c - ci) (c - C2). Let us
now mark the two points Ai and A^ of the z axis whose coordinates are Ci and Cj.
Theee two points lie on the same side of the origin ; and if we suppose, as is
always allowable, that r and t are positive, they lie on the positive part of the
f axis. If the given point A (0, 0, c) lies outside the segment A1A2, A(c) is
negative, and the distance OA is a maximum or a minimum. In order to see
which of the two it is we must consider the sign of 1 - cr. This coefficient
does not vanish except when c = 1 /r ; and this value of c lies between Ci and Cj,
since A (1 /r) = 8'^/r^. But, f or c = 0, 1 — cr is positive ; hence 1 — cr is posi-
tive, and the distance OA is a minimum if the point A and the origin lie on
the same side of the segment A1A2. On the other hand, the distance OA in a
maximum if the point A and the origin lie on different sides of that segment.
When the point A lies between Ai and A2 the distance is neither a minimum
nor a maximum. The case where A lies at one of the points Ai^ A^ is left in
doubt.
Second case. Let s* — ri > 0. One of the two roots Ci and C2 of A (c) = is
positive and the other is negative, and the origin lies between the two points
Ai and At. If the point A does not lie between Ai and A2, A(c) is positive
and there is neither a maximum nor a minimum. If A lies between Ai and
A%^ A(c) is negative, 1 — cr is positive, and hence the distance OA is a minimum.
Third case. Let a^ -rt = 0. Then A (c) = {r + t) {c - Ci), and it is easily
seen, as above, that the distance OA is a minimum if the point A and the origin
lie on the same side of the point ^1, whose coordinates are (0, 0, Ci), and that
there is neither a maximum nor a minimum if the point Ai lies between the point
A and the orighi.
The points Ai and A2 are of fundamental importance in the study of curva-
tnre; they are the principal centers of curvature of the surface S at the point O.
61. MiTimfl and minima of implicit functions. We often need to find
the maxima and minima of a function of several variables which
are connected by one or more relations. Let us consider, for
example, a function a> =/(«, y, z, it) of the four variables Xy y, «, w,
which themselves satisfy the two equations
/i (x, y, «, u) = 0, /,(x, y, «, u) = 0.
For definiteness, let us think of x and y as the independent vari-
ablei, and of z and u as functions of x and y defined by these equa-
tion!. Then the necessary conditions that <u have an extremum are
UI,i61] SINGULAB POINTS MAXIMA AND MIKIMA 129
dx dz dx du dx ' dy dz dy du dy *
and the partial derivatives dz/dxj du/dx, dz/dy, du/dy are given
by the relations
dx dz dx du dx * dx dz dx du dx *
dy dz dy du dy * dy dz dy du dy
The elimination of dz/dx, du/dx, dz/dy^ du/dy leads to the new
equations of condition
'^^ ^ /)(x, z, u) ' Z>(y, z, u) ^'
which, together with the relations /j = 0, /j = 0, determine the val-
ues of a*, y, ;?, w, which may correspond to extrema. But the equa-
tions (27) express the condition that we can find values of X and /i
which satisfy the equations
(28)
dx dx dx ' dy dy dy *
^« ^« "^ <7« du du '^ c?u
hence the two equations (27) may be replaced by the four equations
(28), where A and /i are unknown auxiliary functions.
The proof of the general theorem is self-evident, and we may
state the following practical rule :
Given a function
/(a-i, a-j, . • • , X,)
of n variables, connected by h distinct relations
<^i = 0, <^,=:0 , <^A = 0;
in order to find the values of x^, x^, ••, x^ xrhich may render this
function an extremum xce must equate to zero the partial derivatives
of the auxilia ry function
f -\- K<^\ -r • • -\- K4>k}
regarding Ai, A3, • • , X^ <w constants.
180 TAYLOR'S SERIES [III, §62
M. Aa«thtr example. We shall now take up another example, where the mini-
mam is not necessarily given by equating the partial derivatives to zero. Given
a triangle ABC ; let us try to find a point P of the plane for which the sum
PA + PB + PC of the distances from P to the vertices of the triangle is a
minimnm. Let (ci, 61), (oa, 6a), (08, bs) be respectively the coordinates of the
▼ertices A,B,C referred to a system of rectangular coordinates. Then the func-
tion whose minimum is sought is
(29) « = V(x- ai)a -\-{y- 61)2 +V(x- 03)2 + (y-ft2)^+V(x - as)^+{y -bs)^,
where each of the three radicals is to be taken with the positive sign. This equa-
tion (29) represents a surface S which is evidently entirely above the xy plane,
and the whole question reduces to that of finding the point on this surface which
is nearest the xy plane. From the relation (29) we find
9t
:
X-Oi
-ai)* + (y-
y-bi
-ai)^ + (y-
X -
y-
-aa)2
-Oa
+ {y-
-62
+ (y-
-62)2
-1
X - as
ax
— =
4-
-a8)2 + (y-
y -bi
-hr
«F
V(x-
- a8)2 + {y -
-w
and it is evident that these derivatives are continuous, except in the neighbor-
hood of the points A, B, C, where they become indeterminate. The surface 5,
therefore, has three singular points which project into the vertices of the given
triangle. The minimum of 2 is given by a point on the surface where the tan-
gent plane is parallel to the xy plane, or else by one of these singular points. In
order to solve the equations dz/dx = 0, dz/dy = 0, let us write them in the
form
X — a\ X — Oj X — az
V(x - 01)2 + (y - 6i)2 V(x - aa)2 + (y - 62)^ V(x - a,)2+ (y - 63)2'
y-bi y- 62 y -bz
V'(»-ai)» + (y-6i)2 V(x - 03)2 + (y - 63)2 V(x - 03)2 + (y - 63)2*
Then squaring and adding, we find the condition
1 \ 2 _(^«i)(^ - 02) 4- (y - 61) (y - &2)__ ^ ^
V(x - ai)2 + (y - 6i)2 V(x - 02)2 + (y - 62)^
The geometrical interpretation of this result is easy : denoting by a and p the
eosines of the angles which the direction PA makes with the axes of x and y,
respectively, and by a' and ^ the cosines of the angles which PB makes with the
axes, we may write this last condition in the form
H-2(aa' + /3/S0=O,
or, denoting the angle APB by w,
2 cos w + 1 = 0.
Bmm the condition in question expresses that the segment AB subtends an
•agte of 120* at the point /*. For the same reason each of the angles BPC and
OPA must be 120*».* It is clear that the point P must lie inside the triangle
• The reader is urged to draw the figure.
III,i63] SINGULAR POINTS MAXIMA AND MINIMA 131
ABC, and that there Lb no point which poMOMon the required property if any
angle of the triangle ABC is equal to or greater than 120^. In case none of the
angles is as great as 120°, the point P is uniquely determined by an easy ood«
struction, as the intersection of two circles. In this case the minimum \m gbeo
by the point P or by one of the vertices of the triangle. But it is easy to show
that the sum PA + PB + PC is less than the sum of two of the sides of the tri-
angle. For, since the angles APB and APC are each 120°, we find, from the
two triangles PAC and PBA, the formulse
AB- Va^ + b^ + od, ilC= Va«Tl«Tac,
where PA = a, PB = 6, PC = c. But it is evident that
2 2
and hence
AB-\- AOa-^b + c.
The point P therefore actually corresponds to a minimum.
When one of the angles of the triangle ABC is efjual to or greater than 120"
there exists no point at which each of the sides of the triangle ABC subtends an
angle of 120°, and hence the surface S has no tangent plane which is parallel to
the zy plane. In this case the minimum must be given by one of the vertices of
the triangle, and it is evident, in fact, that this is the vertex of the obtuse angle.
It is easy to verify this fact geometrically.
68. D'Alembert's theorem. Let F(x, y) be a polynomial in the two variables
X and y arranged into homogeneous groups of ascending order
^(x, y) = H + <Pp{x, y) 4- 0p + i(a:, y) + • • • + ^«(x, y),
where fl" is a constant. If the equation <t>p (x, y) — 0, considered as an equation
in y/x, has a simple root, the function F(x, y) cannot have a maximum or a mini-
mum for X = J/ = 0. For it results from the discussion alx)ve that there exist sec-
tions of the surface z + H = F(x, y) made by planes through the z axis, some
of which lie above the xy plane and others below it near the origin. From this
remark a demonstration of d'Alembert's theorem may be deduced. For, let/(z)
be an integral polynomial of degree m,
/(z) = ^0 -I- Axz + Atz* + • • . + An,z^,
where the coefficients are entirely arbitrary. In order to separate the real and
imaginary parts let us write this in the form
/(x -f- xy) = oo + t6o + (ai + i6i) (X + ty) + • • • + (cu + «>»,) (z + <y)",
where oo, bo, ai, &i, - • , ou, hm are real. We have then
f{z) = P-\-iq,
where P and Q have the following meanings :
P = Oo + aix - feiv + • • • ,
Q = bo + 6ix + ail/ + • • . ;
and hence, finally,
|/(«)|=Vp«+<?.
X32 TAYLOR'S SERIES [HI, §63
We will firtt show that I/(z)|, or, what amounts to the same thing, that
P« + Q", cannot be at a minimum for x = y = except when a© = 60 = 0. For
this purpose we shall introduce polar coordinates p and 0, and we shall suppose,
for the sake of generality, that the first coefficient after Aq which does not
Ttnish is Ap. Then we may write the equations
P = Oo + (0,. cosp0 - 6p sinp0)p'' + • • • ,
Q = 60 + (ftp CO8P0 -}- Op sin p0)pP + ••• ,
pt+Qa = a; + 6; + 2pP [(ooOp + 60M cosjx^ + {bottp - aobp) ain p<f>] + • • • ,
where the terms not written down are of degree higher than p with respect to p.
But the equation
(ooOp + bobp) co8p<f> + (bottp - Oobp) 8mp<f> =
gives tanp0 = X", which determines p straight lines which are separated by
angles each equal to 2 it /p. It is therefore impossible by the above remark that
P« + Q8 should have a minimum for x = y = unless the quantities
OoOp + bobpy boOp — Oobp
both vanish. But, since aj + 6J is not zero, this would require that ao = 60 = ;
that is, that the real and the imaginary parts of f{z) should both vanish at the
origin.
If \/{z) I has a minimum for x = or, y = ^, the discussion may be reduced to
the preceding by setting 2 = a + i/3 + 2'. It follows that \f{z)\ cannot be at a
minimum unless P and Q vanish separately f or x = or, y = /3.
The absolute value of f{z) must pass through a minimum for at least one
value of z, for it increases indefinitely as the absolute value of z increases indefi-
nitely. In fact, we have
where the terms omitted are of degree less than 2 m in p. This equation may be
written In the form
Vp^-^(^ = p'n(y/a^ + bl + e),
where t approaches zero as p increases indefinitel y. Henc e a circle may be
drawn whose radius iJ is so large that the value of Vp* + Q2 jg greater at every
point of the circumference than it is at the origin, for example. It follows that
there is at least one point
X = a, y = /3
this circle for which vP« -f Q3 is at a minimum. By the above it fol-
that the point x = a, y = /3 is a point of intersection of the two curves
P = 0, Q = 0, which amounts to saying that 2 = a + /Siis a root of the equation
/(s)=0.
In this example, as In the preceding, we have assumed that a function of the
two variables x and y which is continuous in the interior of a limited region
actually a ss nme s a minimum value inside or on the boundary of that region.
This It a iUtement which will be readily granted, and, moreover, it will be
rigorously demonstrated a little later (Chapter VI).
Ill, Exs] EXERCISES 133
EXERCISES
1. Show that the number tf, which occurs In Lagrange*! form of the re-
mainder, approaches the limit l/(n + 2) ajs A approaches zero, provided that
/('• + «;(a) ia not zero.
2. Let F{x) be a determinant of order n, all of whoee element! are funetiona
of X. Show that the derivative F\x) is the sum of the n determinants obtained
by replacing, successively, all of the elements of a single line by their deriva-
tives. State the corresponding theorem for derivatives of higher order.
3. Find the maximum and the minimum values of the distance from a fixed
point to a plane or a skew curve ; between two variable points on two curves ;
between two variable points on two surfaces.
4. The points of a surface S for which the sum of the squares of the dis-
tances from n fixed points is an extremum are the feet of the normals let fall
upon the surface from the center of mean distances of the given n fixed points.
5. Of all the quadrilaterals which can be formed from four given sides, Uiat
which is inscriptible in a circle has the greatest area. State the analogous
theorem for polygons of n sides.
6. Find the maximum volume of a rectangular parallelepiped inscribed in
an ellipsoid.
7. Find the axes of a central quadric from the consideration that the vertices
are the points from which the distance to the center is an extremum.
8. Solve the analogous problem for the axes of a central section of an ellipaoid.
9. Find the ellipse of minimum area which passes through the three vertices
of a given triangle, and the ellipsoid of minimum volume which passes through
the four vertices of a given tetrahedron.
10. Find the point from which the sum of the distances to two given straight
lines and the distance to a given point is a minimum.
[Joseph BaaTRAKD.]
11. Prov6 the following formulas :
log(z + 2) = 2 log(x + 1) - 2 log (X - 1) -f log(x - 2)
[BoROA*8 Series.]
log(i + 6) = log(x + 4) + log(x + 3) - 2 logx
+ log(x - 3) + log(x - 4) - log(x - 6)
af ^^ iV ^^ V, 1
L«*-26*« + 72 8Vx»-«6x« + 72/ J
[Habo^s Seriea.]
CHAPTER IV
DEFINITE INTEGRALS
I. SPECIAL METHODS OF QUADRATURE
64- Quadrature of the parabola. The determination of the area
bounded by a plane curve is a problem which has always engaged
the genius of geometricians. Among the examples which have
come down to us from the ancients one of the most celebrated is
Archimedes' quadrature of the parabola. We shall proceed to
indicate his method.
Let us try to find the area bounded by the arc A CB of a parabola
and the chord AB. Draw the diameter CD, joining the middle
point D of AB tx> the point C, where the tangent is parallel to AB.
Connect AC and BC, and let E and E' be the points where the
tangent is parallel to BC and
AC J respectively. We shall
first compare the area of the
triangle BEC, for instance,
with that of the triangle ABC.
Draw the tangent ET, which
cuts CD at T. Draw the diam-
eter EF, which cuts CB Sit F;
and, finally, draw EK and FH
parallel to the chord AB. By
an elementary property of the
parabola TC = CK. Moreover,
CT =:EF = KH, and hence
EF=CH/2=CD/4:. The
areas of the two triangles BCE
and BCD, since they have the
•ame base BC, are to each other as their altitudes, or as EF is
to CD. Henoe the area of the triangle BCE is one fourth the area
of the triangle BCD, or one eighth of the area ^^ of the triangle ABC.
The area of the triangle A CE' is evidently the same. Carrying out
the tame process upon eacH of the chords BE, CE, CE', E'A, we
134
Fio.8
IV,§d5j
SPECIAL METHODS
135
obtain four new triangles, the area of each of which is 5/8*, and so
forth. The nth operation gives rise to 2" triangles, each having the
area S/S*. The area of the segment of the parabola is evidently
the limit approached by the sum of the areas of all these triangles
as n increases indefinitely ; that is, the sum of the following descend-
ing geometrical progression :
S S
-f--
and this sum is 4 S/3. It follows that the required area ijt equal to
two thirds of the area of a parallelogram whose sides are AB and CD.
Although this method possesses admirable ingenuity, it must be
ailmitt*jd that its success dejxinds essentially upon certain special
properties of the parabola, and that it is lacking in generality. The
other examples of quadratures which we might quote from ancient
writers would only go to corroborate this remark : each new curve
required some new device. But whatever the device, the area to be
evaluated was always split up into elements the number of which
was made to increase indefinitely, and it was necessary to evaluate
the limit of the sum of these partial areas. Omitting any further
particular cases,* we will proceed at once to give a general method
of subdivision, which will lead us naturally to the Integral Calculus.
65. General method. For the sake of definiteness, let us try to
evaluate the area S bounded by a curvilinear arc AM By an axis xx'
which does not cut that arc, and two perpendiculars ^.lo a-^d BB^ let
fall upon xx' from y
the points .^1 and B.
We will suppose
further that a par-
allel to these lines
AA^y BBq cannot
cut the arc in more
than one point, as
indicated in Fig. 9.
Let us divide the segment A^Bq into a certain number of etjual or
unequal parts by the points P|, Pj, •••, P,_i, and through these
points let us draw lines PiQi, I\Qt^ •••, P«-iQ«_i parallel to AA^
and meeting the arc AB in the points Qiy Q„ •••, Q,_„ respectively.
x'o
^^
Rt
fti
A, />, A
/Vi5»
FiQ. 9
* A lar^o number of examples of determinations of areas, arcs, and Tolomas by
the methods of ancient writers are to be found in Duhamel's TraUi,
186 DEFINITE INTEGRALS [IV, §65
Now draw through A a line parallel to xx', cutting Pi Qi at q^ ;
through Qi a parallel to xx', cutting P^Q^ at qij and so on. We
obtain in this way a sequence of rectangles Ri, R^, • • • , /?„ • • • , /?„.
Each of these rectangles may lie entirely inside the contour ABBqAq,
but some of them may lie partially outside that contour, as is
indicated in the figure.
Let a< denote the area of the rectangle Ri, and )S, the area bounded
by the contour Pi.xPiQiQi-i- In the first place, each of the ratios
fii/aif pt/aty •••, Pi/ocij ••• approaches unity as the number of
points of division increases indefinitely, if at the same time each
of the distances A^Pu P1P2, ■" , Pt-iPi, •• approaches zero. For
the ratio ft/a,, for example, evidently lies between li/Pi_iQi_i and
L{/P{,iQi_i, where Z,- and Z,- are respectively the minimum and the
maximum distances from a point of the arc Q,- _ 1 Q, to the axis xx'.
But it is clear that these two fractions each approach unity as the
distance Pi^xPi approaches zero. It therefore follows that the ratio
A + A + --- + ft
which lies between the largest and the least of the ratios cti/fSi,
^i/fiit •'} ^n/Pni will also approach unity as the number of the
rectangles is thus indefinitely increased. But the denominator of
this ratio is constant and is equal to the required area S. Hence
this area is also equal to the limit of the sum «! + org H + ^n> 3,s
the number of rectangles n is indefinitely increased in the manner
specified above.
In order to deduce from this result an analytical expression for
the area, let the curve AB be referred to a system of rectangular
axes, the x axis Ox coinciding with xx', and let y =/(«) be the
equation of the curve AB. The function /(a:) is, by hypothesis, a
continuous function of x between the limits a and b, the abscissae
of the points A and B. Denoting hy x^, x^, ■ - • , x^_i the abscissae
of the points of division Pj, P,, -.., P,._i, the bases of the above
rectangles are «i - a, x, - sci, . . . , jb^ - a;.._i, • . • , i - x„_^, and their
altitudes are, in like manner, f{a),f(x,), ..., f{x,_,), ••., f{x^_,).
Henoe the area S is equal to the limit of the following sum :
(1) (x» - a)/(a) + (x, - x,)f{x,) + • • • + (6 - a:,_0/(a:„_0,
M the number n increases indefinitely in such a way that each of
the differences a^ ~ a, a-j - a?i, ... approaches zero.
IV. {t»] 8P£C1AL MKTUODb 187
66. Examples. If the base AB be divided into n equal parts, each
of length h {b ^ a = nh), all the rectangles have the same base h,
and their altitudes are, respectively,
/(«), /(« + A), /(a + 2 A), . . ., /[a + (n - 1) A].
It only remains to find the limit of the sum
A }/(«) -f /(a + h) +f(a + 2 A) -h • • • +/[a + (n - 1)A] j,
where
as the integer n increases indefinitely. This calculation becomes
easy if we know how to find the sum of a set of values /(x) corre-
sponding to a set of values of x which form an arithmetic progres-
sion; such is the case if f(x) is simply an integral power of x, or,
again, iff(x)= sin7rtx or/(x)= coswix, etc.
Let us reconsider, for example, the parabola x* = 2pi/, and let us try
to find the area enclosed by an arc OA of this parabola, the axis of x,
and the straight line x = a which passes through the extremity A.
The length being divided into n equal parts of length A (nA = a), we
must try to find by the above the limit of the sum
2^[A» + 4A« + . .. + («-!)• A']=g[l +4 + 9 + ■•. +(»-!)•].
The quantity inside the parenthesis is the sum of the squares of the
first (n — 1) integers, that is, 7i(n — 1) (2 n — l)/6; and hence the
foregoing sum is equal to
n(n-l)(2n~l)
12 pn*
As n increases indefinitely this sum evidently approaches the limit
a*/6p = (l/3){a. a^/2p)y or one third of the rectangle constructed
upon the two coordinates of the point Ay which is in harmony with
the result found above.
In other cases, as in the following example, which is due to
Format, it is better to choose as points of division points whose
abscissae are in geometric progression.
Let us try to find the area enclosed by the curve y ^ Ax»^, the
axis of X, and the two straight lines x a a, x = 6 (0 < a < 6), where
188 ' DEFINITE INTEGRALS [IV, §66
the exponent fi is arbitrary. In order to do so let us insert between
a and *, n — 1 geometric means so as to obtain the sequence
a, a(l + a), a(l + «)S -", «(1 + «)""'> *>
where the number a satisfies the condition a (1 + «)" = *• Tak-
ing this set of numbers as the abscissae of the points of division, the
corresponding ordinates have, respectively, the following values :
and the area of the ^th rectangle is
Hence the sum of the areas of all the rectangles is
Aa'^'^'all + (1 + ay + ' + (1 + a)»f'* + i)+ •••+(! + «y»-i)<'^ + ^>].
If ^ -f 1 is not zero, as we shall suppose first, the sum inside the
parenthesis is equal to
or, replacing a (1 -f- a)* by b, the original sum may be written in the
form
a
^(JM + l_aM + l)
(l-^ay + '-l
As a approaches zero the quotient [(1 H- a)**"^^ — l]/a approaches
as its limit the derivative of (1 + a^-^^ with respect to a for a = 0,
that is, fi + l'y hence the required area is
/* + !
If ;i = — 1, this calculation no longer applies. The sum of the
areas of the inscribed rectangles is equal to nAa, and we have to
find the limit of the product na where n and a are connected by the
relation
a(l + a)* = 6.
From thifl it follows that
na — log - z — r = log -
''alog(l -Ha) *a
log(l + ay
IV.JflT]
sprrrvT. ^fETHODS
189
where the symbol log den
approaches zero, (1 4- a;'/* a|
uct na approaches log (6 /a).
A\og(b/a).
"' (perian logarithm. As a
i<* number «, aiid the prod-
Hence the required area U equal to
Fio. 10
67. Primitive functions. The invention of the Integral Calculus
reduced the proljltiii of evaluating a plane area to the problem of
finding a function whose derivative is known. Let y =/(af) be the
equation of a curve referred to two rectangular axes, where the
function f(x) is continuous. Let us consider the area enclosed by
this curve, the axis of ar, a fixed ordinate 3/o/*o> and a variable
ordinate MP, as a function of the abscissa x of the variable ordinate.
In order to include all pos-
sible cases let us agree to
denote by A the sum of the
areas enclosed by the given
curve, the x axis, and the
straight lines MqPoj ^^f*f
each of the portions of
this area being affected
by a certain sign: the
sign -f- for the portions to
the right of MqPq and above Ox, the sign — for the portions to the
right of 3/yPo and below Ox, and the opposite convention for por-
tions to the left of MqJ^q. Thus, if MP were in the position M'P', we
would take A equal to the difference
MoPoC - M'P'C;
and likewise, if MP were at M"P", A = M"P"D - MoPoD.
With these conventions we shall now show that the derivative of
the continuous function A, defined in this way, is precisely f(x). As
in the figure, let us take two neighboring ordinates MP, XQ, whose
abscissae are x and x -f- Ax. The increment of the area A.4 evidently
lies between the areas of the two rectangles which have the same
base PQ, and whose altitudes are, respectively, the greatest and the
least ordinates of the arc MN. Denoting the maximum ordinate by
// and the minimum by A, we may therefore write
AAx < A^ < ^Ax,
or, dividing by Aj, h < A.I /Ax < //. As Ax approaches zero, FT and
h aj)pruiu.'h the same limit MP, or /(x), since /(x) is continuous.
140 DEFINITE INTEGRALS [IV, §G8
Hence the derivative of A is /(a;). The proof that the same result
holds for any position of the point M is left to the reader.
If we already know a primitive function oif{x), that is, a function
F(x) whose derivative i8/(x), the difference A — F(x) is a constant,
since its derivative is zero (§ 8). In order to determine this con-
stant, we need only notice that the area A is zero for the abscissa
a; = a of the line MP. Hence
A =F(x)-F(a).
It follows from the above reasoning, first, that the determination
of a plane area may be reduced to the discovery of a primitive func-
tion; and, secondly (and this is of far greater importance for us),
that every continuous function f(x) is the derivative of some other
function. This fundamental theorem is proved here by means of
a somewhat vague geometrical concept, — that of the area under a
plane curve. This demonstration was regarded as satisfactory for a
long time, but it can no longer be accepted. In order to have a stable
foundation for the Integral Calculus it is imperative that this theo-
rem should be given a purely analytic demonstration which does not
rely upon any geometrical intuition whatever. In giving the above
geometrical proof the motive was not wholly its historical interest,
however, for it furnishes us with the essential analytic argument of
the new proof. It is, in fact, the study of precisely such sums as
(1) and sums of a slightly more general character which will be
of preponderant importance. Before taking up this study we must
first consider certain questions regarding the general properties of
functions and in particular of continuous functions.*
II. DEFINITE INTEGRALS ALLIED GEOMETRICAL CONCEPTS
68. Upper and lower limits. An assemblage of numbers is said to
have an upper limit (see ftn., p. 91) if there exists a number N so
large that no member of the assemblage exceeds N. Likewise, an
assemblage is said to have a lower limit if a number N' exists than
which no member of the assemblage is smaller. Thus the assem-
blage of all positive integers has a lower limit, but no upper limit j
• Among the most important works on the general notion of the definite integral
tlMM ■hould be mentioned the memoir by Rieraann : Vher die Mdglichkeit, eine Func-
tion dureh eine trigonometrische Reihe darzustellen (Werke, 2d ed., Leipzig, 1892,
p.SS0; and alio French translation by Ijiugel, p. 226) ; and the memoir by Dferboux, to
wbleh we have already referrod : Sur lea fonctionn ilitcontinueft {Annuleti de VEcole
Normals Bup^rieure, 2d seriMt, Vol. IV).
IV,§.wj ALLIED GEOMETRICAL CONCEPTS 141
the assemblage of all integers, positive aiid negatiye, has neither;
and the assemblage of all rational numbers between and 1 has
both a lower and an upper limit.
Let (E) be an assemblage which has an upper limit With
respect to this assemblage all numbers may be divided into two
classes. We shall say that a nunilxT a belongs to the first class if
there are memlx*rs of the assemblage (E) which are greater than a,
and that it belongs to the second class if there is no member of the
assemblage (/;) greater than a. Since the assemblage (E) has an
upper limit, it is clear that numbers of each class exist. If A he
a number of the first class and B a number of the second class, it
is evident that A <^; there exist members of the assemblage (E)
which lie between A and B^ but there is no member of the assem-
blage (E) which is greater than B. The number C = (v4 -f B)/2
may belong to the first or to the second class. In the former case
we should replace the interval (A, B) by the interval (C, /i), in the
latter case by the interval (/I, C). The new interval (Ai^ B^) is half
the interval (.1, B) and has the same properties : there exists at least
one member of the assemblage (A') which is greater than .-li, but none
which is greater than Bi. Operating upon (.4i, Bi) in the same way
that we operated upon (A, B), and so on indefinitely, we obtain an
unlimited sequence of intervals (.1, J5), (Ai, Bi), (/I,, 5j), •••, each
of which is half the preceding and possesses the same property
as (.1, B) with respect to the assemblage (E). Since the numbers
.1, A I, vlj, •••, A^ never decrease and are always less than iJ, they
approach a limit X (§ 1). Likewise, since the numl)ers By B^ 5,, • •
never increase and are always greater than A , they approach a limit X'.
Moreover, since the difference B^ — .1, =(^B — -4)/2" approaches zero
as n increases indefinitely, these limits must be equal, i.e. X' = X.
Let L be this common limit; tlien L is called the vjiper limit of the
assemblage (A'). From the manner in which we have obtained it,
it is clear that L has the following two properties :
1) No tnemher of the assemblage (E) w greater than L.
2) There alirat/s exists a member of the assembla/je (E) which is
ijreater than Z, — e, where c is any arbitrarily small positive number.
For let us suppose that there were a member of the assemblage
greater than Z., say L + A (A > 0). Since B^ approaches L as n
increases indefinitely, B^ will be less than L -^ h after a certain
value of 71. But this is imjx>ssible since B^ is of the second class.
On the other hand, let c be any positive numl)er. Then, after a
142 DEFINITE INTEGRALS [IV, §69
certain value of n, i4. will be greater than i — € ; and since there are
members of (E) greater than A^, these numbers will also be greater
tlum L — €. It is evident that the two properties stated above can-
not apply to any other number than L.
The upper limit may or may not belong to the assemblage (E).
In the assemblage of all rational numbers which do not exceed 2,
for instance, the number 2 is precisely the upper limit, and it belongs
to the assemblage. On the other hand, the assemblage of all irra-
tional numbers which do not exceed 2 has the upper limit 2, but
this upper limit is not a member of the assemblage. It should be
particularly noted that if the upper limit L does not belong to the
assemblage, there are always an infinite number of members of (E)
which are greater than L — c, no matter how small c be taken. For if
there were only a finite number, the upper limit would be the largest
of these and not L. When the assemblage consists of n different
numbers the upper limit is simply the largest of these n numbers.
It may be shown in like manner that there exists a number L\
in case the assemblage has a lower limit, which has the following
two properties :
1) No member of the assemhlage is less than L'.
2) There exists a member of the assemblage which is less than
V 4- c, where t is an arbitrary positive number.*
This number Z' is called the lower limit of the assemblage.
69. Oscillation. Let f(x) be a function of x defined in the closed f
interval (a, b) ; that is, to each value of x between a and b and to each
of the limits a and b themselves there corresponds a uniquely deter-
mined value of /(x). The function is said to he finite in this closed
interval if all the values which it assumes lie between two fixed
numbers A and B. Then the assemblage of values of the function
has an upper and a lower limit. Let M and m be the upper and
lower limits of this assemblage, respectively ; then the difference
•Whenerer ail numbers can be separated into two classes A and B, according to
any charactarlitic property, in such a way that any number of the class .4 is less than
any number of the claM B, the upper limit L of the numbers of the class A is at the
mm» time the lower limit of the numbers of the class B. It is clear, first of all, that
any nttmbw greater than L l)elong8 to the class B. And if there were a number L'<L
; to thaolaM B, then every number greater than // would belong to the class B,
&twj numb«r lew than L belongs to the class A, every number greater than L
balonfi to the claas B, and L itself may belong to either of the two classes.
t Tha word " doMd " to used merely for emphasis. See § 2. — Trans.
IV, §70] ALLIED GEOMETRICAL CONCEPTS 143
A s= Af — m is called the oscillation of the function /(x) in the
interval (a, b).
These definitions lead to sevcial ifinarks. In order that a funo-
tion be finite in a closed interval i^u, b) it is not sufficient that it
should have a finite value for every value of x. Thus the function
defined in the closed interval (0, 1) as follows :
/(O) = 0, /(x) = 1/x for X > 0,
has a finite value for each value of x ; but nevertheless it is not
finite in the sense in which we have defined the word, for/(x) > A
if we take x<l/A. Again, a function which is finite in the closed
interval (a, b) may take on values which differ as little as we please
from the upper limit M or from the lower limit m and still never
assume these values themselves. For instance, the function /(x),
defined in the closed interval (0, 1) by the relations
/(0) = 0, f(x) = l-x for 0<x<l,
has the upper limit M = 1, but never reaches that limit.
70. Properties of continuous functions. We shall now turn to the
study of continuous functions in particular.
Theorem A. Letf(x) be a function which is continunus ui mr nosed
interval (a, b) and € an arbitrary positive number. Then we can
always break up the interval (a, b) into a certain number of partial
intervals in such a way that for any two' values of the variable
whatever^ a:' a7id x", which belong to the same partial interval^ we
always have |/(x')— /(x")| < c.
Suppose that this were not true. Then let c=(a-|-d)/2; at
least one of the intervals (a, c), (c, b) would have the same prop-
erty as (rt, 6); that is, it would be impossible to break it up into
partial intervals which would satisfy the statement of the theorem.
Substituting it for the given interval (a, b) and carrying out the
reasoning as above (§ 68), we could form an infinite sequence of
intervals (a, ft), (oj, ft,), (a„ 6,), • •, each of which is half the preced-
ing and has the same property as the original interval (a, b). For
any value of n we could always find in the interval (o., b^) two
numbers x' and x" such that (/(x')— /(x")| would be larger than c.
Now let X be the common limit of the two sequences of numbers
a, ai, aj, • •• and b, bi, /^,, • • •. Since the function /(j-) is continuous
for X = X, we can find a number i; such that |/(x)— /(X)| <€/2
144 DEFINITE INTEGRALS [IV, §70
wheneyer |x - A| is less than 17. Let us choose n so large that both
a, and d, differ from k by less than rj. Then the interval (a„, K)
will lie wholly within the interval (X - 1;, X + ^7) ; and if x' and x"
are any two values whatever in the interval (a„, 6„), we must have
\f(x') -/(X) I < c/2, \f(x").-f(\) I < e/2,
and hence l/Cx*) -/(x") | < e. It follows that the hypothesis made
above leads to a contradiction ; hence the theorem is proved.
Corollary L Let a, Xj, ar„ • • • , x^.i, i be a method of subdivision
of the interval (a, h) into p subintervals, which satisfies the con-
ditions of the theorem. In the interval (a, x^ we shall have
|/(x) I < |/(a) I + e ; and, in particular, \f(xy) \ < \f{a) \ + c. Like-
wise, in the interval (xi, x^ we shall have |/(ic)| < |/(a:i) | -f- c,
and, a fortioriy \f(x) \ < \f(a) | 4- 2 c ; in particular, for x = x^,
|/(^) I < |/(") I + 2 € ; and so forth. For the last interval we shall
have
l/WI< 1/(^,-01 + '<!/« I +i'«-
Hence the absolute value of f(x) in the interval (a, b) always
remains less than |/(a) 1 4- p^. It follows that every function which
is continuous in a closed interval (a, b) is finite in that interval.
Corollary II. Let us suppose the interval (a, b) split up intojo sub-
intervals (a, Xx)j (xj, Xj), • • • , (Xp_i, b) such that |/(x') —f(x")\< c/2
for any two values of x which belong to the same closed subinterval.
Let 17 be a positive number less than any of the differences x^ — a,
X, — Xj • • • , 6 — Xp_i. Then let us take any two numbers whatever
in the interval (a, b) for which |x' — x"| < 77, and let us try to find
an upper limit for |/(x') — /(x")|. If the two numbers x' and x"
fall in the same subinterval, we shall have |/(x') — /(x")|<c/2.
If they do not, x' and x" must lie in two consecutive intervals,
and it is easy to see that |/(x') -f(x") \ < 2 (e/2) = e. Hence cor-
responding to any positive member e another positive number -q can be
found such that
|/(x')-/(x")|<.,
where x' and x" are any two numbers of the interval (a, b) for which
Ix* — »"|<iy. This property is also expressed by saying that the
function f(x) is uniformly continuous in the interval (a, b).
Thborbm B. a function f(x) which is continuous in a closed
intemal (a, b) takes on every value between f(a) and f(b) at least
once for some value of x which lies between a and b.
IV, §70] ALLIED GEOMETRICAL CONCEPTS 146
Let us first consider a particular case. Suppose that /(a) and
f{b) have opposite signs, — that /(a) < and /(A) > 0, for instance.
We shall then show that thero exists at least one value of x between
a and b for which f{x) = 0. Now/(3r) is negative near a and posi-
tive near b. Let us consider the assemblage of values of x between
a and b for which f{x) is positive, and let A be the lower limit of
this assemblage (a < X < b). By the very definition of a lower
limit /(A — h) is negative or zero for every positive value of h.
Hence /(A), which is the limit of /(A — A), is also negative or zero.
But /(A) cannot be negative. For suppose that /(A) = — m, where
m is a positive number. Since the function f(x) is continuous for
X = A, a number rj can be found such that \f(x) — /(A)| < m when-
ever |a; — A| < 17, and the function f(x) would be negative for all
values of x between A and A -h 17. Hence A could not be the lower
limit of the values of x for which /(x) is positive. Consequently
/(X) = 0.
Now let N be any number between f(a) and f(b). Then the
function <^(x) =f(x) — N is continuous and has opposite signs for
X = a and x = b. Hence, by the particular case just treated, it
vanishes at least once in the interval (a, b).
Theorem C. Every function which w continuous in a closed inter-
val (a, b) actnally assumes the value of its upper and of its lower
limit at lea^t once.
In the first place, every continuous function, since we have
already proved that it is finite, has an upper limit M arid a lower
limit m. Let us show, for instance, that /(x) = 3/ for at least one
value of xin the interval (<«, b).
Taking c = (a •\- b)/2, the upper limit of f(x) is equal to M for
at least one of the intervals (a, c), (<•, 6). Let us replace (a, b)
by this new interval, repeat tlie process upon it, and so forth.
Reasoning as we have already done several times, we could form
an infinite sequence of intervals (a, b)y (a^, 6,), (a,, A,), ••., each of
which is half the preceding and in each of which the upper limit of
/(x) is M. Then, if A is the cx)mmon limit of the sequences a, Ci,
• • • , a„ • • • and A, ij, • • • , ft,, • • • , /(A) is equal to M. For suppose that
/(A) = M — hf where h is positive. We can find a positive number
rj such that/(x) remains between /(A) 4- A/ 2 and /(A) — A/ 2, and
therefore less than A/— A/2 as long as x remains between A — 17
and A -I- 17. Let us now choose n so great that a„ and ft, differ from
their common limit A by less than rj. Then the interval (a,, ft,) lies
146 DEFINITE INTEGRALS [IV, §71
wholly inside the interval (\ — rj,\ + -q), and it follows at once
that the upper limit of f{x) in the interval (a„, b^ could not be
equal to 3/.
Combining this theorem with the preceding, we see that any func-
tion which is continuous in a closed interval (a, h) assumes, at least
oneey every value between its upper and its lower limit. Moreover
theorem A may be stated as follows : Given a function which is
continuous in a closed interval (a, b), it is possible to divide the inter-
val into such small subregions that the oscillation of the function in
any one of them will be less than an arbitrarily assigned positive
number. For the oscillation of a continuous function is equal to
the difference of the values of /(a) for two particular values of the
variable.
71. The sums S and s. Let f(x) be a finite function, continuous
or discontinuous, in the interval (a, b), where a<b. Let us sup-
pose the interval (a, b) divided into a number of smaller partial
intervals (a, Xi), (xi, Xg), •••, (ajp-i, b), where each of the numbers
a; J, a;,, • • •, Xp.j is greater than the preceding. Let M and m be the
limits of f{x) in the original interval, and M^ and m, the limits
in the interval («,_!, Xf), and let us set
S = M^(x^ - a) 4- M,(x^ -x^)^--- + M^(b - x^_,),
s = mi (Xi — a) -{- m2(x2 — x{) -] h m^ (b — iCp_i).
To every method of division of (a, b) into smaller intervals there
corresponds a sum S and a smaller sum s. It is evident that none
of the sums S are less than m(b — a), for none of the numbers Af^
are less than m ; hence these sums S have a lower limit /.* Like-
wise, the sums «, none of which exceed M(b — a) have an upper
limit r. We proceed to show that /' is at most equal to I. For this
purpose it is evidently sufficient to show that s<S' and s' ^ .S", where
5, s and 5', s' are the two sets of sums which correspond to any
two given methods of subdivision of the interval (a, b).
In the first place, let us suppose each of the subintervals (a, Xj),
('i» *,), ••• redivided into still smaller intervals by new points of
division and let
<h Vu y»f "i t/k-it a^i, yjt+i, •., yj_i, Xj, 2// + i, ••, b
• ll/(«) itaeoiuiUnt, 8=a,M=:m, and, in general, all the inequalities mentioned
equAtiou. — TaANi.
IV, §72] ALLIED GEOMETRICAL CONCEPTS UJ
be the new suite thus obtained. This new method of subdivision
is called consecutive to the first. Let S and <r denote the sums anal-
ogous to .S' and s with respect to this new method of division of the
interval (a, b), and let us compare S and « with 2 and o-. Let us
compare, for example, the portions of the two sums S and 2 which
arise from the interval (a, Xi). Let M[ and m[ be the limits of
f(x) in the interval (a, y,), 3/,' and m^ the limits in the interval
(i/i) yi)i ' ■ ) ^^k ^^^ W the limits in the interval (jft-u *i)* Then
the portion of S which comes from (a, x^ is
M{{y, - «) + A/Ky« - y,) + • • + 3/i(ar, - y,_0 ;
and since the numbers 3/|, A/j, •• •, 3// cannot exceed 3/,, it is clear
that the above sum is at most equal to 3/i (a*i — a). Likewise, the
portion of 2 which arises from the interval (x„ a*,) is at most equal
to 3/,(a'j — a-,), and so on. Adding all these inequalities, we find
that 2 ^ ^y and it is easy to show in like manner that <r ^ ».
Let us now consider any two methods of subdivision whatever,
and let Sy s and S\ s' be the corresponding sums. Superimposing
the points of division of these two methods of sulxlivision, we get a
third method of subdivision, which may be considered as consecu-
tive to either of the two given methods. Let 2 and <r be the sums
with respect to this auxiliary division. By the above we have the
relations
2<5, «r>«, 2<.S% (r>»';
and, since 2 is not less than o-, it follows that a'< 5 and 8<S', Since
none of the sums S are less than any of the sums «, the limit /
cannot be. less than the limit /'; that is, /^/'.
72. Integrable functions. A function which is finite in an inter-
val (a, b) is said to l>e mtegrable in that interval if the two sums
S and s approac^h the same limit when the number of the partial
intervals is indefinitely increased in such a way that eacli ^f t>iocp
partial intervals approaches zero.
The necessary and sufficient condition that a function he integrable
in an interval is that corresponding to any positive number c another
nuviber rj exists sttch that S — s is less than c whenever each of the
partial intervals is less than rj.
This condition is, first, necessary, for if &' and s have the same
limit /, we can find a number ij so small that |5 ~ /"I wd |« — /| are
148 DEFINITE INTEGRALS [IV, §72
each less than c/2 whenever each of the partial intervals is less
than iy. Then, a fortiori, S — sis less than c.
Moreover the condition is sufficient, for we may write *
S-s = S-I + I-I' + I'-s,
and since none of the numbers S - I, I - I', I' - s can be negative,
each of them must be less than c if their sum is to be less than c.
But since / — /' is a fixed number and c is an arbitrary positive
number, it follows that we must have /' = /. Moreover S — I<€
and I — s<€ whenever each of the partial intervals is less than rj,
which is equivalent to saying that S and s have the same limit /.
The function f(x) is then said to be integrahle in the interval
(a, b), and the limit / is called a definite integral. It is represented
by the symbol
I=^Jf(x)dx,
which suggests its origin, and which is read ''the definite integral
from a to 6 of f(x) dx.'' By its very definition / always lies between
the two sums S and s for any method of subdivision whatever.
If any number between S and s be taken as an approximate value
of /, the error never exceeds S — s.
Every continuous function is integrahle.
The difference 5 — s is less than or equal to (h — a)ui, where
a» denotes the upper limit of the oscillation of f(x) in the partial
intervals. But ri may be so chosen that the oscillation is less than
a preassigned positive number in any interval less than r} (§ 70).
If Uien i; be so chosen that the oscillation is less than e/(b — a),
the difference S — s will be less than «.
Any monotonically increasing or monotonically decreasing function
in an interval is integrahle in that interval^
A function /(x) is said to increase rnonotonically in a given interval
(a, b) if for any two values x\ x" in that interval f(x') ^.f(x") when-
ever x' > x". The function may be constant in certain portions of the
interval, but if it is not constant it must increase with x. Dividing
the interval (a, b) into n subintervals, each less than rj, we may write
S =/(xO (X, - a) +/(x,) (X, - xO + • • • +/(ft) (b - x_0,
M ^f{a)(x, - a) +/(xO(x. - xO -\""+f(x,,,)(b - x,.,),
•For Um proof that / and r exist, see §73, which may be read before § 72. —Trans.
1V,J721 ALLIED GEOMETRICAL CONXEPTS 149
for the upper limit of f(x) in the interval (a, arj), for instance,
is precisely /(x^), the lower limit /(a); and so on for the other
subintervals. Hence, subtracting,
i- - * = (X, - o)[/(x,)-/(a).l + (X. - x,)[/(x.) -/(x,)]
+ •• -l-(*-x...)[/(«)-/(x._,)].
None of the differences which occur in the right-hand side of this
equation are negative, and all of the differences Xi — o, x^ — x^
'■• are less than 17; consequently
S-»< 1) [/('.) -/(«) +/{'.) -/(^.) + • • • +/(i) - /(=".-.)].
or
5-,<,[/(i)-/(a)],
and we need only take
in order to make S — s< t. The reasoning is the same for a mono-
tonically decreasing function.
Let us return to the general case. In the definition of the inte-
gral the sums S and s may be replaced by more general expres-
sions. Given any method of subdivision of the interval (a, b) :
a, a:i, a;,, • •, ar,_,, ar,, ••., x^_^, b;
let ^1, ^2» • • , ^,, • • • be values belonging to these intervals in order
(a-,_, < $i < Xi). Then the sum
(2)|X/(ft)(^^-^.-i) =
I' V(^i)(^i - «) +/(«(^« - xi) + • • • +f($nKb - *.-,)
evidently lies between the sums S and », for we always have
7/1, </(^,) < 3/,. If the function is integrable, this new sum has the
limit /. In particular, if we suppose that ^1, ^„ • • , ^, coincide
with o, Xj, ••., «,_„ respectively, the sum (2) reduces to the sum
(1) considered above (§ C>5).
There are several propositions which result immediately from the
definition of the integral. We have supposed that a < ft ; if we now
interchange these two limits a and 6, each of the factors X| — «,_,
changes sign ; hence
J j\x)dx=.-jyix)dx.
150 DEFINITE INTEGRALS [IV, §72
It also evidently follows from the definition that
rf(x)dx = ff(x)dx 4- ff(x)dx,
%/a *Ja Jc
at least if c lies between a and h\ the same formula still holds when
h lies between a and c, for instance, provided that the function f{x)
is integrable between a and c, for it may be written in the form
ff(x)dx = ff(x)dx - ff(x)dx = ff(x)dx + rf(x)dx.
If f{x) = A<l>(x) + B\l/(x), where A and B are any two constants,
we have
fix)dx = A I <f>(x)dx-\- B I xlf(x)dx,
and a similar formula holds for the sum of any number of functions.
The expression /(^,) in (2) may be replaced by a still more gen-
eral expression. The interval (a, h) being divided into n sub-
intervals (a, Xi), • • • , (x,_i, a;,), • • • , let us associate with each of the
subintervals a quantity ^„ which approaches zero with the length
05, — x,_i of the subinterval in question. We shall say that ^^
approaches zero uniformly if corresponding to every positive num-
ber c another positive number t^ can be found independent of i and
such that l^;^ I < c whenever x^ — Xi_^ is less than iy. We shall now
proceed to show that the sum
«'=X[/(^i-.)+ «('».■-»=.•-.)
(=1
approaches the definite integral ]^f(x)dx as its limit provided
that ^, approaches zero uniformly. For suppose that 17 is a number
80 small that the two inequalities
X/(^<-i)(aJ.-aj,-0- ^f{x)dx
<e, 1^,1 <C
are satisfied whenever each of the subintervals ic» — a;,._i is less
than 17. Then we may write
'-£'
/{x)dx =
[X/(»<-i)(a'< - 'i-x) -£nx)dx'\ + ^ i,{x, - x,_o,
IV. 573] ALLIED GEOMKTKICAL CONCEPTS 151
and it is clear that we shall have
-f.
f{x)dx
<€ + «(*-«)
whenever each of the subintervals is less than rj. Thus the theorem
is proved.*
78. Darboux't theorem. Given any function /(z) which \& finite in an inter-
val (a, b)\ tlie sums S and s approach their limitii / and /% respectively, when
tlie number of subintervals increases indefinitely in such a way that each of
them approachee zero. Let ua prove this for the sum »S', for instance. We
shall suppose that a<6, and that/(z) is positive in the interval (a, 6), which can
be brought about by adding a suitable constant to/(x), which, in turn, amounts
to adding a constant to eacli of the sums S. Then, since the number / is the
lower limit of all the sums ^, we can find a particuhir method of subdivision, say
for which the sum S is less than I + f/2, where e is a preassigned positive num-
ber. Let us now consider a division of (a, b) into intervals less than ij, and let us
try to find an upper limit of the corresponding sum S'. Taking first those inter-
vals which do not include any of the points Zi, x^, • • •, 2p_i, and recalling the
reasoning of § 71, it is clear that the portion of iS" which cumes from these inter-
vals will be less than the original sum S, that is, less than / -f e/2. On the other
hand, the number of intervals which include a point of the set Zj, Zt, • • • , Xp-\
cannot exceed p — 1, and hence their contribution to the sum S' cannot exceed
(p — \)Mri, where if is the upper limit of /(z). Hence
S'<7-|-e/2 + (p-l)Jfi»,
and we need only choose ij less than e/2 3f (p - 1) in order to make S' less than
/ + «. Hence the lower limit / of all the sums S is also the limit of any sequence
of 5's which corresponds to uniformly infinitesimal subintervals.
It may be shown in a similar manner that the sums a have the limit /'.
If the function /(z) is any function whatever, these two limits I and /' are in
general different. In order that the function be integrable it is necessary and
sufficient that /' = I.
74. First law of the mean for integrals. From now on we shall
assume, unless something is explicitly said to the contrary, that
the functions under the integral sign are continuous.
* The al)ove theorem can be extended without difficulty to double and triple inte-
gralH ; we shall make use of it in several places ($§ 8i), 96, 97, 131, 144, etc.).
The pniposition is essentially only an application of a theorem of Dobamel's
at^*ortiinK to which the limit of a sum of inHnitesimals remains unchanged when
each of tht> infiiiitcsimalH is replactnl hy another infinitesimal which differs from the
given JntiniteHinml by an iiitinittwinml of higher order. (See an article hy W. P.
Osgtxxl, AnnaU of Matheuiatic*, '31 series, Vol. IV, pp. 161-178: Th9 Inteffral <u
the JAinit of a Sum ami a Thtoretn o/ Duhamel'a.)
152 DEFINITE INTEGRALS [IV, §74
Let/(x) and ^(x) be two functions which are each continuous
in the interval (a, b), one of which, say <t>(^), has the same sign
throughout the interval. And we shall suppose further, for the
sake of definiteness, that a<b and <f> (x) > 0.
Suppose the interval (a, b) divided into subintervals, .and let
i it •••» A» ••• ^ values of x which belong to each of these
smaller intervals in order. All the quantities f(i,) lie between the
limits M and m of /(x) in the interval (a, b) :
m<f(i,)<M.
Let us multiply each of these inequalities by the factors
respectively, which are all positive by hypothesis, and then add
them together. The sum ^f($^)<|>($i)(Xi-x^_^) evidently lies
between the two sums w2<^(^,) (x.- - x,._i) and M2</)(^.) (x,- - x,_,).
Hence, as the number of subintervals increases indefinitely, we
have, in the limit,
/
f <f>(x)dx^ I f(x)<ji(x)dx<M j <l>(x)dx,
a J a Jo-
which may be written
fix) <t>(x)dx = ix j <f>(x) dxj
where /i lies between m and M. Since the function f{x) is con-
tinuous, it assumes the value /i for some value ^ of the variable
which lies between a and b ; and hence we may write the preceding
equation in the form
(3) rf(x)<t>(x)dx =/(^) C\{x)dx,
%Ja %J a
where i lies between a and b. * If, in particular, <^ (x) = 1 , the
integral ^ dx reduces to {b — a) by the very definition of an inte-
gral| and the formula becomes
w £f(x)dx=(b
«)/(«)•
• Tha lower ilgn hoIdH in the preceding relations only when/(i) = k. It is evidenc
that iIm) furmula «iUl holdit, however, and that a<( < & in any case. —Trans.
IV, § 75] ALLIED GEOMETRICAL CONCEPTS 154
76. Second law of tho mean for integrals. There is a second formula, due to
Bonnet, which he deduced from an Important lemma of AbePs.
Lemma. Let co, <i, • • ,9pbeaHiof monoUmicaUy decrmutng pogUive quamU-
ties^ anduo,ui,- • ,UpOie »ame number of arbitrary positive or negative quaniUiee.
If A and B are rtapectively the greatest and tke leatt qf cUl cfthe nune fo ss ««,
s, = uo -F uj . • • • , 1^ = Wo + ui + • • • + «p, <A« turn
8 = «oiio + nwi + • • • + fpH^
will lie between Ato and JUo, i.e. Ato ^ >' ^ IUq.
For we have
whence the sum S is equal to
«o(<o - <i) + ai (<i - <j) + • -f ap-i (<p-i - «p) + tp€p.
Since none of the differences <o — <i, <i — <«, • • •, <p-i — «i» are negatiye, two
limits for S are given by replacing Soi Si , • • • , «p by their upper limit A and then
by their lower limit B. In this way we find
8<A{«o-€i-\-ei-et + \- ep_i - ep -f ep) = Ate,
and it is likewise evident that »S' ^ Be©.
Now let/(x) and ^(x) be two continuous functions of z, one of which, ^(«),
is a positive monotonically decreasing function in the interval a<x<b. Then
tlje integral f^f{x)<p(x)dx is the limit of the sum
/(a)0(a)(xi - a) +/(zi)0(xi)(xa - xi) + • • ••
The numbers 0(a), 4>{xi), - • • form a set of monotonically decreasing positive
numbers; hence the above sum, by the lemma, lies between A<p(a) and B<p{a)t
where A and B are respectively the greatest and the least among the following
sums:
/(a)(xi-a),
/(a) {xi - a) + /(xi) (X, - xi) ,
/(a) (xi - a) +/(xi) (X, - xi) + . . . +/(x,-i) (6 - «,_i).
Passing to the limit, it is clear that the integral in question must lie between
Ai^{a) and Bi(p((t), where Ai and Bi denote the maximum and the minimum,
respectively, of the integral f^^/{x)d£, as c varies from a to 6. Since this inte-
gral is evidently a continuous function of its upper limit c (§ 76), we may write
the following formula :
(6) j^/(x)0(x)dx = 0(a)j]/(x)dx, a<(<b.
When the function ^(x) is a monotonically decreasing function, without
being always positive, there exists a more general formula, due to Weientrasa.
In such a case let us set ^ (x) = (b) + f (x). Then f (x) is a positiTe monotoo-
ically decreasing function. Applying the formula (6) to it, we find
jj(x)^{x)dx = [0(a) - 0(6)] j] /(x)dx.
154 DEFINITE INTEGRALS [IV, §76
From this it is easy to derive the formtda
J /(x)0(x)dx =^ /(x)0(6)dx + [0(a) - 0(6)] j^ f{x)dx,
or ^
y /(x)0(x)dx = if>{a)fjf{x)dx-]- <f>{b)f f{x)dx.
SimilAr formulte exist for the case when the function 0(x) is increasing.
76. Return to primitive functions. We are now in a position to
give a purely analytic proof of the fundamental existence theorem
(S 67). hetJXx) be any continuous function. Then the definite integral
^<*>=X
f(t)dt,
where the limit a is regarded as fixed, is a function of the upper
limit X. We proceed to show that the derivative of this function
isf{x). In the first place, we have
Xx + h
f(t)dt,
or, applying the first law of the mean (4),
F{x^h)-F(x) = hf{t),
where ^ lies between x and x -\- h. As /i approaches zero, /(^)
approaches f{x) ; hence the derivative of the function F(x) is f{x),
which was to be proved.
All other functions which have this same derivative are given
by adding an arbitrary constant C to F(x). There is one such
function, and only one, which assumes a preassigned value yQ for
« = a, namely, the function
yo+jJ{t)dt.
When there is no reason to fear ambiguity the same letter x is
uaed to denote the upper limit and the variable of integration, and
f'f(x)dx is written in place of fjf{t)dt. But it is evident that
a definite integral depends only upon the limits of integration and
the form of the function under the sign of integration. The letter
which denotes the variable of integration is absolutely immaterial.
Every function whose derivative is f(x) is called an indefinite
inUffral of /(x), or a jyrimitive function of f{x)y and is represented
by tiie symbol
IV, fW] ALLIED GEOMETRICAL CONCEPTS 166
the limits not being indicated. By the above we evidently have
Conversely, if a fuuction F{x) whose derivative is /(x) can be
discovt^rt'd bv :iiiv iiietliod wliatever, we mav write
i
f{x)dx = F(ar)-H C.
In order to determine the constant C we need only note that the
left-hand side vanishes for x = a. Hence C = — /^(a), and the
fundamental formula becomes
(6) £/{x)dx = F(x)-F{a).
If in this formula /(j-) be replaced by F'(ar), it becomes
F(x)-F(a)=£FXx)dx,
or, applying the first law of the mean for integrals,
F(x)-F(a) = (x-a)F'(0,
where $ lies between a and x. This constitutes a new proof of the
law of the mean for derivatives ; but it is less general than the one
given in section 8, for it is assumed here that the derivative F'(x) is
continuous.'
We shall consider in the next chapter the simpler classes of func-
tions whose primitives are known. Just now we will merely state
a few of those which are apparent at once :
Ja(x - aydx = A ^^~2^l^' + ^> a -h 1 :9fc 0;
/
A — — = A log (x — a)'\- C;
I cosxf/r =r sin J- 4- c'; j s'mxdx = ^ COBX -^ C;
f
eT'dx = — + r, m ^ Oj
]£6 DEFINITE INTEGRALS [IV, {76
rj^ = arctana + C; J;^^= = arosina! + C;
J-^= = log(» + V?TT)+C, J=a^ = log/(x)+C.
The proof of the fundamental formula (6) was based upon the
assumption that the function f{x) was continuous in the closed inter-
val (a, b). If this condition be disregarded, results may be obtained
which are paradoxical. Taking f{x) = l/x% for instance, the for-
mula (6) gives
'dx 1 1
b
rdx_
Ja ^'~
The left-hand side of this equality has no meaning in our present
system unless a and b have the same sign ; but the right-hand side
has a perfectly determinate value, even when a and b have different
signs. We shall find the explanation of this paradox later in the
study of definite integrals taken between imaginary limits.
Similarly, the formula (6) leads to the equation
If /(a) and/(6) have opposite signs, /(x) vanishes between a and b,
and neither side of the above equality has any meaning for us at
present. We shall find later the signification which it is convenient
to give them.
Again, the formula (6) may lead to ambiguity. Thus, if
f(x)^l/(l + x^), we find
f dx
arc tan b — arc tan a.
Here the left-hand side is perfectly determinate, while the right-
hand side has an infinite number of determinations. To avoid this
ambiguity, let us consider the function
'<-'-X'r^.
This function F(x) is continuous in the whole interval and van-
iihet with x. Let us denote by arc tan x, on the other hand, an
angle between - 'w/2 and + 7r/2. These two functions have the
IV, $77] ALLIED GEOMETRICAL CONCEPTS 167
same derivative and they both vanish for x = 0. It follows thai
they are equal, and we may write the equality
r' dx r'_dx_ rdx_ ^ .
I r-— -5 = I ^ . , — I r-; — i = arc tan A - arc tan a,
where the value to be assigned the arctangent always lies between
-7r/2 and 4-7r/2.
In a similar manner we may derive the formula
/^ = arc sin b — arc sin a,
VI — x'
i:
where the radical is to be taken positive, where a and b each lie
l)etween — 1 and -f 1, and where arc sin x denotes an angle which
lies between — 7r/2 and -f 7r/2.
77. Indices. In general, when the primitive F{x) is multiply determinate, we
ghould choose one of the initial values F{a) and follow the continuous variation
of this brancli as x varies from a to b. Let us consider, for instance, the integral
J. p'+v 1 1 +/'(»)
where
/(X) = ^
and where P and Q are two functions which are both continuous in the interval
(a, b) and which do not both vanish at the same time. If Q does not vanish
between a and 6, /(x) does not become infinite, and arc tan/(x) remains between
— n/2 and + ic/2. But this is no longer true, in general, if the equation Q =
has roots in this interval. In order to see how the formula must be modified, let
us retain the Convention that arc tan signifies an angle between — ic/2 and -|- x/2,
and let us suppose, in the first place, that Q vanishes just once between a and 6
for a value x = c. We may write the integral in the form
r V(^)dx r^- . r'""'4. r*
where e and e' are two very small positive numbers. Since /(«) does not become
Infinite between a and c - c, nor between c -i- 1' and 6, this m«y again be written
X
6
-^-— = arc tan/(c - c) - arc tAn/(a)
+ arc tan/(6) - arc t«n/(c + O + f
Several cases may now present themselves. Suppose, for the sake of definite-
ness, that /(x) becomes infinite by passing from + » to — eo. Then /(c — t) will
be positive and very large, and arc tan/(c - c) will be very near to it/2 ; while
158 DEFINITE INTEGRALS [IV, § 78
tit + O w»» ^ negative and very large, and arc tan/(c + O will be very near
- ir/2. Alao, the integral /^If will be very small in absolute value; and,
to the limit, we obtain the formula
X'
£
Zi?)*L =^ + arctan/(6) - arctan/(a).
f„ l+/«(x)
Similarly, it is eaay to show that it would be necessary to s^traxt tc if /(x)
pMsed from - oo to + 00. In the general case we would divide the interval
(a, 6) into subintervals in such a way that /(x) would become infinite just once
in each of them. Treating each of these subintervals in the above manner and
adding the resulta obtamed, we should find the formula
' r(x)dg ^ arctan/(6) - arctan/(a) ^ {K - K') it,
where K denotes the number of times that/(x) becomes infinite by passing from
+ 00 to - 00, and K' the number of times that f{x) passes from — co to + «.
The number K - K' is called the index of the function /(x) between a and 6.
When/(x) reduces to a rational function Vi/V, this index may be calculated
by elementary processes without knowing the roots of V. It is clear that we
may suppose Vi prime to and of less degree than F, for the removal of a poly-
nomial does not affect the index. Let us then consider the series of divisions
necessary to determine the greatest common divisor of Fand Fi, the sign of the
remainder being changed each time. First, we would divide F by Fi, obtaining
a quotient Qi and a remainder - Fa. Then we would divide Vi by F2, obtaining a
quotient Q» and a remainder — Fs ; and so on. Finally we should obtain a con-
stant remainder - F«+ 1. These operations give the following set of equations :
F = FiQi - Fa,
Vl =F2Q2-F8,
F„_i = F„Q„-Fn+i.
The sequence of polynomials
(7) F, Fi, Fa, .-., Vr-u Fr, Fr+i, ..-, F„, F„+i
has the essential characteristics of a Sturm sequence : 1) two consecutive poly-
nomials of the sequence cannot vanish simultaneously, for if they did, it could
be shown successively that this value of x would cause all the other polynomials
to vanish, in particular Fn + i; 2) when one of the intermediate polynomials Fi,
Ft, • • • , Vn vanishes, the number of changes of sign in the series (7) is not altered,
lor if Vr vanishes for x = c, Vr-i and Fr + i have different signs for x = c. It
follows that the number of changes of sign in the series (7) remains the same,
eioept when z passes through a root of F = 0. If Fi/ F passes from + 00 to — oo,
this number increases by one, but it diminishes by one on the other hand if
Vi/V passes from - 00 to +00. Hence the index Is equal to the difference of
the number of changes of sign in the series (7) for x = 6 and x = a.
78. Aret of a curve. We can now give a purely analytic definition
of the area bounded by a continuous plane curve, the area of the
notangle only being considered known. For this purpose we need
IV. §78] ALLIED GEOMETRICAL CONCEPTS 159
only translate into geometrical language the results of $ 72. Let
f(x) be a function which is continuous in the closed interval (a, 6),
and let us suppose for definiteness that a<b and that fix) > in
the interval. Let us consider, as above (Fig. 9, % 65), the portion of
the plane bounded by the contour AMBBqA^^ composed of the seg-
ment .-io/iu of the X axis, the straight lines AA^ and BB^ parallel to
tlie y axis, and having the abscissa; a and 6, and the arc of the curve
A MB whose ecjuation is y =f(x). Let us mark off on -.^o^o a certain
number of points of division Pj, Pj, • • , Pi^i, Pa "t whose abscissa.'
are a-,, Xj, • ••, a;^.,, a*,, •••, and through these points let us draw
parallels to the y axis which meet the arc A MB in the points
Qu Qtf •••» Qi-iy Qi) •••> respectively. Let us then consider, in
particular, the portion of the plane bounded by the contour
Qi-iQiPiPi~iQi-ii and let us mark upon the arc Q,_iQ.- the highest
and the lowest points, that is, the points which correspond to the
maximum 3/,- and to the minimum w,. of f(x) in the interval
(ir,_,, Xf). (In the figure the lowest point coincides with Q,_i.)
Let /ij be the area of the rectangle Pf_i /',*,«._ i erected upon the
base Pi. I Pi with the altitude 3/„ and let r,- be the area of the
rectangle Pi-iPiQiQi-i erected upon tlie base Pi^iPi with the alti-
tude 7/1,. Then we have
Ri = Mi(Xi - ar,_,), r^ = m,.(a;< - x,_,),
and the results found above (§ 72) may now be stated as follows :
whatever be the points of division, there exists a fixed number /
which is always less than 2/^, and greater than 2r,, and the two
sums 2/f,- and 2r, approach / as the number of sabintervals P<_i/*i
increases in such a way that each of them approaches zero. We shall
call this common limit / of the two sums 2/?, and 2r, the area of
the portion of the plane hounded by the contour AMBBqA^A. Thus
the area under consideration is defined to be equal to the definite
integral j^f{x)dx.
This definition agrees with the ordinary notion of the area of a
plane curve. For one of the clearest points of this rather vague
notion is that the area bounded by the contour P<_i/*<<2,n<Q<_,/*,_,
lies Ixjtween the two areas /?< and r, of the two rectangles P,_iP,*,«,-.i
and P,-i/*,7,Q,_i; hence the total area bounded by the contour
AMBBqAqA must surely be a quantity which lies between the two
sums 2/?, and 2r,. But the definite integral / is the •nly fixed quan-
tity which always lies between these two sums for any mode of
subdivision of A^B^y since it is the common limit of 2/^^ and 2r<.
160
DEFINITE INTEGRALS
[IV, §79
The given area may also be defined in an infinite number of other
ways as the limit of a sum of rectangles. Thus we have seen that
the definite integral / is also the limit of the sum
2(^,-aj,_i)/(«,
where ^, is any value whatever in the interval (ic,_i, x,). But the
element
(x.-^i-,)/(f)
of this sum represents the area of a rectangle whose base is Pi^iPi
and whose altitude is the ordinate of any point of the arc Qi.itiiQi.
It should be noticed also that the definite integral / represents
the area, whatever be the position of the arc A MB with respect to
the X axis, provided that we adopt the convention made in § 67.
Every definite integral therefore represents an area ; hence the calcu-
lation of such an integral is called a quadrature.
The notion of area thus having been made rigorous once for all,
there remains no reason why it should not be used in certain
arguments which it renders nearly intuitive. For instance, it is
perfectly clear that the area considered above lies between the areas
of the two rectangles which have the common base AqBq, and which
have the least and the greatest of the ordinates of the arc A MB,
respectively, as their altitudes. It is therefore equal to the area of
a rectangle whose base is AqBq and whose altitude is the ordinate
of a properly chosen point upon the arc A MB, — which is a restate-
ment of the first law of the mean for integrals.
79. The following remark is also important. Let /(a;) be a func-
tion which is finite in the interval (a, b) and which is discontinuous
in the manner described below for
a finite number of values between
a and b. Let us suppose that f(x)
is continuous from c to c-\-k(k>0),
and that /(c -}- c) approaches a cer-
tain limit, which we shall denote
by f(c + 0), as e approaches zero
through positive values; and like-
wise let us suppose that f(x) is
continuous between c — k and c and that f(c — e) approaches a limit
/(« — 0) as c approaches zero through positive values. If the two
limits f(r. + 0) and /(o - 0) are different, the function f(x) is dis-
continuoufl for x = e. It is usually agreed to take for /(c) the
IV, 5 80] ALLIED GEOMETRICAL CONCEPTS 161
value [f(c + 0) -\-f{e - 0)]/2. If the function /(z) has a certain
number of points of discontinuity of this kind, it will be repre-
sented graphically by several distinct arcs AC^ CD, D'B. Let o
and d, for example, be the abscissa) of the points of discontinuity.
Then we shall write
f(x)dx = / f(x)dx -f / f{x)dx + f f{x)dx,
«/• */«r Jd
in accordance with the definitions of § 72. Geometrically, this taninie
integral represents the area bounded by the contour A CC'IjD'HHqAqA.
If the upper limit b now be replaced by the variable x, the definite
integral
n^)=jj<^
x^dx
is still a continuous function of x. In a point x where f(x) is con-
tinuous we still have F'(x) = f(x). For a point of discontinuity,
X = c for example, we shall have
Xe + h
f(x) dx = hf(c + Bh), < d < 1,
and the ratio [F(c -h h)— F(c)'\/h approaches /(c -f 0) or/(r — 0)
according as h is positive or negative. This is an example of a
function F(x) whose derivative has two distinct values for certain
values of the variable.
80. Length of a curvilinear arc. Given a curvilinear arc AB; let us
take a certain number of intermediate points on this arc, m^, w,,
•'> ''^n-i » ^"d let us construct the broken line ylmjm, •• • m,_,B by
connecting each pair of consecutive points by a straight line.
If the length of the perimeter of this broken line approaches a
limit as the number of sides increases in such a way that each of
them approaches zero, this limit is defined to be the length of the
arc AB.
Let
be the rectangular coordinates of a point of the .uv Ati expressed
in terms of a parameter t, and let us suppose that as / varies from
a to b{a<b) the functions /, ^, and ^ are continuous and possess
continuous first derivatives, and that the point (x, y, x) describes
the arc AB without changing the sense of its motion. Let
162 DEFINITE INTEGRALS [IV, §80
be the values of t which correspond to the vertices of the broken
Una Then the side c, is given by the formula
or, applying the law of the mean to «,• — aj^^i, • •,
where ^,, 17^ C.- li© between ^,_i and ^f. When the interval {ti_^, i^)
is very small the radical differs very little from the expression
v[/'(«.-.)]' + [-^'(«.-i)P + [fc'.-or-
In order to estimate the error we may write it in the form
But we have
and consequently
/^fe)+/fe-0
<1.
Hence, if each of the intervals be made so small that the oscillation
of each of the functions f'(t), <f>'(t), \l/'(t) is less than e/3 in any
interval, we shall have
where
h.i<«i
and the perimeter of the broken line is therefore equal to
The supplementary term 2€,(^< — ^^.i) is less in absolute value
than €l(ti — ^<_,), that is, than €(* — a). Since € may be taken as
small as we please, provided that the intervals be taken sufficiently
•mall, it follows that this term approaches zero ; hence the length S
of the arc i4B is equal to the definite integral
(8) S= C V/'a + <^'^ + .A" dt.
This definition may be extended to the case where the derivatives
/, ^', ^' are discontinuous in a finite number of points of the arc AB,
IV, §80] ALLIED GEOMETRICAL CONCEPTS 168
which occurs when the curve has one or more comers. We need only
tlivide the arc AB into several parts for each of which/', ^', ^' are
continuous.
It results from the formula (8) that the length S of the arc
between a fixed point A and a variable point M, which corresponds
to a value t of the parameter, is a function of t whose derivative is
whence, squaring and multiplying by eft*, we find the formula
(9) dS^ = rfx'* 4- rfy* + dz^,
which does not involve the independent variable. It is also easily
remembered from its geometrical meaning, for it means that dS is
the diagonal of a rectangular parallelopiped whose adjacent edges are
dXf di/j dz.
Note. Applying the first law of the mean for integrals to the
definite integral which represents the arc MqMh whose extremities
correspond to the values t^^ t^ of the parameter (tx > t^)^ we find
* = arc M^I, = (t, - to) ^f\e) -f <t>'\e) 4- ^'\e),
where lies in the interval (^„, ti). On the other hand, denoting
the chord ^l^i^Il by c, we have
Applying the law of the mean for derivatives to each of the differ-
ences /(^i) — /(^»), • •, we obtain the formula
c==(h- to) Vf'\$) -h ^'\f,) -f ^'\0y
where the three numbers ^, 17, ( belong to the interval (t^t <i). By
the above calculation the difference of the two radicals is less than c,
provided that tlie oscillation of each of the functions/'(<), ^'(0» ^'(0
is less than </3 in the interval (^o* U)- Consequently we have
»-c<c(<, - to),
or, finally,
If the arc ^fa^fl is infinitesimal, <t~ '0 approaches zero; hence c,
iuid therefore also 1 — r/s, approaches aero. It follows that ths ratio
of an injinitesinuil arc to its chord approaches unity as its limit.
164 DEFINITE INTEGRALS [IV, §81
ExampU. Let us find the length of an arc of a plane curve whose
equation in polar coordinates is p =/(w). Taking w as independent
variable, the curve is represented by the three equations a; = p cos w,
y = p sin o», ;e = ; hence
da* = dx^ -f rfy* =(cos o) rfp — p sin <u d<tif + (sin mdp -\- p cos w dmy,
or, simplifying,
ds^ = dp^'{-p''dio\
Let us consider, for instance, the cardioid, whose equation is
p = R -\- Rcos w.
By the preceding formula we have
ds^ = R^dw^ [sin^o) + (1 + cos a>)2] = 4:R^ cos^ | cio)*,
or, letting q> vary from to tt only,
ds = 2R cos — c?<u ;
and the length of the arc is
(4..sin|)"\
where u>o and <oi are the polar angles which correspond to the extrem-
ities of the arc. The total length of the curve is therefore 8 R.
81. Direction cosines. In studying the properties of a curve we are
often led to take the arc itself as the independent variable. Let us
choose a certain sense along the curve as positive, and denote by s
the length of the arc AM between a certain fixed point A and a vari-
able point 3/, the sign being taken + or — according as M lies in
the positive or in the negative direction from A. At any point M
of the curve let us take the direction of the tangent which coincides
with the direction in which the arc is increasing, and let a, ^, y be
the angles which this direction makes with the positive directions
of tlie three rectangular axes Ox, Oy, Oz. Then we shall have the
following relations :
cos or _ cos /8 cosy _ 1 ±1
dx " dy ^ dz ^ ^dx'' -f dy^ -f"^ ^ ~di'
To find which sign to take, suppose that the positive direction of
the tangent makes an acute angle with the x axis ; then x and s
inorea«e simultaneously, and the sign -f should be taken. If the
angle a is obtuse, cos a is negative, x decreases as s increases, dx/ds
IV, $82] ALLIED GEOMETRICAL CONCEFFS IfiS
is negative, and the sign -f should be taken again, iience in any
case the following formula; hold :
(10) C08a=— , COS^ = -^, ^y^^,
where rfx, rfy, rf«, ds are differentials taken with respect to the same
independent variable, which is otherwise arbitrary.
82. Variation of a segment of a straight line. Let AfM^ be a segment
of a straight line whose extremities describe two curves C, f,. On
each of the two curves let us choose a
point as origin and a positive sense of
motion, and let us adopt the follow-
ing notation : «, the arc AM \ «!, the arc
AiMxi — the two arcs being taken with
the same sign ; /, the length MM^ ; ^, the
angle between MM^ and the positive di-
rection of the tangent 3/7'; 6^, the angle
between Mi M and the positive direction
of the tangent M^ 7\. We proceed to
try to find a relation between B, $i and the differentials ds, rf#,, dl.
Let (x, y, «), (jcx, yi, z^) be the coordinates of the points M, Mi,
respectively, a, ft, y the direction angles of M7\ and a,, fii, yj the
direction angles of Mi 7\. Then we have
P = (x-xiy-^(y-yiy-^(z-^ziy,
from which we may derive the formula
Idl = (x -Xi)(dx - dxi) + (y - yx)(dy - dyi) 4- (« - «,)(<& - <i«,),
which, by means of the formulae (10) and the analogous formulas
for 6\, may be written in the form
dl
= (^^C03aH-^^COS/9 + ^^C08y)<i«
-f (^^ cos a, + ^^ cos ft + ^S^ cos y,) rf*,.
But (x — Xi)/l, {y — y\)/li (« — «i)/^ are the direction cosines of
Ml M, and consequently the coefficient of <i5 is ~ cos 6. Likewise
the coefficient of (^i is — cos ^; hence the desired relation is
(10') dl = - ds cos $ - dsioos $1.
We shall make frequent applications of this formula; one such we
proceed to discuss immediately.
166 DEFINITE INTEGRALS [IV, §88
8S. Theorems of Graves and of Chasles. Let E and E' be two confocal ellipses,
aod let the two tangents MA, MB to the interior ellipse E be drawn from a point
3f, which lies on the exterior ellipse E\ The
difference MA + MB — arc ANB remains con-
stant as the point M describes the ellipse E\
Let s and s' denote the arcs OA and OB,
a the arc (XM, I and V the distances AM and
BM, 6 the angle between MB and the positive
direction of the tangent MT. Since the ellipses
are confocal the angle between MA and MT is
p. -„ equal to it — 0. Noting that AM coincides
with the positive direction of the tangent at A,
and that BM is the negative direction of the tangent at B, we find from the
formula (10'), successively,
dl = — ds ■\- dff cos 6 ,
dl' = ds' — d<r cos ;
whence, adding,
d{l+V)=d (8' -s)=d (arc ANB),
which proves the proposition stated above.
The above theorem is due to an English geometrician, Graves. The following
theorem, discovered by Chasles, may be proved in a similar manner. Given an
ellipse and a confocal hyperbola which meets it at N. If from a point M on that
branch of the hyi)erbola which passes through N the two tangents MA and MB
be drawn to the ellipse, the difference of the arcs NA — NB will be equal to the
difference of the tangents MA — MB.
m. CHANGE OF VARIABLE INTEGRATION BY PARTS
A large number of definite integrals which cannot be evaluated
directly yield to the two general processes which we shall discuss
in this section.
84- Change of variable. If in the definite integral ///(«) dx the
variable x be replaced by a new independent variable t by means
of the substitution x = <l>(t), a new definite integral is obtained.
Let U8 suppose that the function <^(^) is continuous and possesses a
continuous derivative between a and jS, and that <^(^) proceeds from
a to 6 without changing sense as t goes from a to p.
The interval (a, P) having been broken up into subintervals by
the intermediate values a, t^, t^, . . ., ^„_,, ^, let a, x^,x^, - • ., £c„_i, b
be the corresponding values of a; = <^(^). Then, by the law of the
mean, we shall have
where $^ lies between f,_, and tf. Let ft •-= <^(d,.) be the corresponding
falue of « which lies between x^^i and a,. Then the sum
IV,}*»J CHANGE OF V A 11 TABLE 167
approaches the given definite integral as its limit But this sum
may also be written
and in this form we see that it approaches the new definite integral
X
/[*(<)]♦'(<)<«
as its limit. This establishes the equality
(U) fAx)dx = /"/[*(<)] *'(0*.
•/a »/«
which is called the foT*mula for the change of variable. It is to
be observed that the new differential under the sign of integration
is obtained by replacing x and dx in the differential /(x)rfx by their
values <^(^) and <f>'{t)dt^ while the new limits of integration are the
values of t which correspond to the old limits. By a suitable choice
of the function <^(^) the new integral may turn out to be easier to
evaluate than the old, but it is impossible to lay down any definite
rules in the matter.
Let us take tlie definite integral
i (x-ay-i-p^'
for instance, and let us make the substitution x = a -{- fit. It
becomes
p
or, returning to the variable x,
1/ x — a ^ ^ a\
7, arc tan — 1- arc tan - 1.
^\ P P/
Not all the hypotheses made in establishing the formula (11) were
necessary. Thus it is not necessary that the function <f>(t) should
always moye in the same sense as t varies from a to fi. For defi-
niteness let us suppose that as t increases from <t to y (y < j8), <f>(t)
steadily increases from a to c (c>b)\ then as t increases from y to
fi, <f>(t) decreases from c to b. If the function /(«) is continuous in
the interval (a, r), the formula may be applied to each of the inter-
vals (a, c), (r*, l})y whicli gives
168 DEFINITE INTEGRALS tIV,§84
or, adding,
fjXx)dx = ( f[.'t>{t)-]^Xt)dt.
On the other hand, it is quite necessary that the function </>(^)
should be uniquely defined for all values of t. If this condition be
disregarded, fallacies may arise. For instance, if the formula be
applied to the integral j^^ dxj using the transformation x = ^'^,
we should be led to write
//^=/
\-Jtdt,
which is evidently incorrect, since the second integral vanishes. In
order to apply the formula correctly we must divide the interval
(— 1, -f 1) into the two intervals (— 1, 0), (0, 1). In the first of
these we should take a; = — V? and let t vary from 1 to 0. In the
second half interval we should take x = V^ and let t vary from
to 1. We then find a correct result, namely
NoU, If the upper limits h and ^ be replaced by x and t in the
formula (11), it becomes
f'jlx)dx=f/ii,(t)2,l,Xt)dt,
%Ja %J a
which shows that the transformation x = <f>(t) carries a function
F(ar), whose derivative is /(x), into a function ^(t) whose derivative
"/[^(O] ^'(0* "^^^^ ^^o follows at once from the formula for the
derivative of a function of a function. Hence we may write, in
general,
jA^)dx^jfi4^{t)-]4>\t)dt,
which is the formula for the change of variable in indefinite
integrals.
IV, iM] INTEGRATION BY PARTS 169
85. Integration by parts. Let u and v be two functions which,
togetlier witli tlieir derivatives u' and v', are continuous between a
and b. Then we have
d(uv) dv . du
— ^ — ^ = u — -f- V — »
dx dx dx
whence, integrating both sides of this equation, we find
Ja dx J^ dx J^ dx
This may be written in the form
(12) / udv = \uv']\- I vdUy
%Ja %Ja
where the symbol \F{^x)\ denotes, in general, the difference
F(i)-F(a).
If we replace the limit 6 by a variable limit x, but keep the limit a
constant, wliich amounts to passing from definite to indefinite inte-
grals, this formula becomes
(13)
I udv = uv — I t' du.
Thus the calculation of the integral J udv is reduced to the cal-
culation of the integral fvdu, which may be easier. Let us try,
for example, to calculate the definite integral
I x^logxcfoj, w -f 1 ^ 0.
Setting u = logx, v = x'"-^^ /(tn -f 1), the formula (12) gives
-[
g"-^'loga; _ af^' T
m-fl (w-fl)*J.'
This formula is not applicable if wt -f- 1 = ; in that particular
case we have
/'.og 4^ =[.; (log ,).];.
It is possible to generalize the formula (12). Let the succes-
sive derivatives of the two functions u and v be represented by
u', u", . ., tt<"+»>; v', w", ••, r<" + »>. Then the application of the
170 DEFINITE INTEGRALS [IV, §85
formula (12) to the integrals /Mrft/"^ /w'<^v^'*"^^ ••• leads to the
following equations:
X6 •»& />&
%Ja *Ja
dx.
Multiplying these equations through by +1 and —1 alternately,
and then adding, we find the formula
(14)
+ (-l)'' + i I u^^'^^^vdxy
which reduces the calculation of the integral Juv^'^'^^^dx to the cal-
culation of the integral fu^^-^^^vdx.
In particular this formula applies when the function under the
integral sign is the product of a polynomial of at most the nth
degree and the derivative of order (n + 1) of a known function v.
For then m<'' + '> = 0, and the second member contains no integral
signs. Suppose, for instance, that we wished to evaluate the definite
integral
£e^'f{x)dx,
where /(x) is a polynomial of degree n. Setting w =f(x), v = e'*'Y<»>""^S
the formula (14) takes the following form after ef^"" has been taken
out as a factor :
The same method, or, what amounts to the same thing, a series of
integrations by parts, enables us to evaluate the definite integrals
J co%mxf(x)dXy j smmxf(x)dx,
where /(«) is a polynomial.
IV, J 8(5] INTEGRATION BY PARTS 171
86. Taylor's series with a remainder. In the formula (14) let us
replace u by a function F(x) which, together with its first n + 1
derivatives, is continuous between a and 6, and let us set v = (b—xy.
Then we have
v' = - n{b - xy-\ v" = n(n - l)(b - x)--«, • • .,
v<-)=(~l)-1.2...n, t;<--^») = 0,
and, noticing that r, v', v", •., «;<■-*) vanish for x=b,we obtain the
following equation from the general formula :
=(- l)"! n!F(^») - n\F(a) - n\F'(a) (b - a)
- ~ F"(a)(6 - ay---- F^^\a)(b - a)-l
-h (- 1)- -^ ' r f <- ^ '\x) (b - xydxy
which leads to the equation
Since the factor (6 — x)" keeps the same sign as x varies from a to
i, we may apply the law of the mean to the integral on the right,
which gives
F"*'\x){b - xydx = f *"+"({) / (* - xydx
where ^ lies between a and b. Substituting this value in the preced-
ing equation, we find again exactly Taylor's formula, with Lagrange's
form of the remainder.
87. TraDScendenUl character of e. From the formula (15) we can prove a
famouK theorem due to llermite : The number e is not a root qf any algebraic
equation whose co^cienta are all intcgertt.*
Setting a = and w = - 1 in the formula (16), it becomes
J^«-«/(«)ci»=-C«-«jr(x)]J,
* The present proof ii doe to D. Hilbert. who drew hir inspiration from the method
used by Hermite.
172 DEFINITE INTEGRALS [IV, §87
idMM
F(x) =f(x) +r{x) + '" +/<''>(aj);
and ibiB again may be written in the form
(16) F(b) = «*F(0) - ^f/{^) e-'^dx.
Now let us suppoee that e were the root of an algebraic equation whose coeflB-
cienu are all integers :
Co + Cie + cae2 + . . . + c^e™ = 0.
TbeUf setting 6 = 0, 1, 2, • • • , m, successively, in the formula (16), and adding
the results obtained, after multiplying them respectively by Co, Ci, • • -, c^, we
obtain the equation
(17) coF(0) + c, ^(1) + . . • + c„ F{m) + ^ CieQ{x)e-^dx = 0,
»=o
where the index t takes on only the integral values 0, 1, 2, • • • , m. We proceed
to show that such a relation is impossible if the polynomial /(x), which is up to
the present arbitrary, be properly chosen.
I«et us choose it as follows :
•^(*) = , ^ ,., ^^~Hx - 1)^(« -2)P"-{x- m)p,
(p-l)l
where p is a prime number greater than m. This polynomial is of degree
mp + p — 1, and all of the coefficients of its successive derivatives past the pth
are integral multiples of p, since the product of p successive integers is divisible
by p!. Moreover /(x), together with its first (p — 1) derivatives, vanishes for
X = 1, 2, . • • , m, and it follows that F{1), F{2), • • • , F(m) are all integral mul-
tiples of p. It only remains to calculate .F(O), that is,
F(0) =/(0) +/'(0) + . . . +f(p-^^0) +/(i')(0) +/(P + i)(0) + . . ..
In the first place, /(O) = /'(O) = . . • =/(p-2)(0) = 0, while /(p)(0), /^^ + i)(0), • • •
are all integral multiples of p, as we have just shown. To find /(p - 1) (0) we need
only multiply the coefficient of xp-i in/(x) by (p - 1) !, which gives ± (1 . 2 • • • m)P.
Hence the sum
CoF(0) + CiF(l) + --- + c«F(m)
b equal to an Integral multiple of p increased by
± Co(l . 2 . • . m)p.
If p be taken greater than either m or Co, the above number cannot be divisible
by p ; hence the first portion of the sum (17) will be an integer different from zero.
We shall now thow that the sum
Vc.c^ r/(x)e-'(te
eao be made ■mailer than any preaaeigned quantity by taking p sufficiently large.
Am z Tariee from to i each factor of /(x) is less than in ; hence we have
IV,}*!!] INTEOUATION BY PARTS 178
'^*'"<(F^rij] "'■'*'-'•
'+#-1.
xc^j;/(x)
IJo''^' r(p-l)! Jo ^(p- 1)1
from which it follows that
e-'dx <jr^__-.e- = 0(p),
where If is an upper limit of |co | + | Ci | + h | c. | • Ae p increaoes indefi-
nitely the function <f> (p) approaches zero, for it is the general term of a conrer-
gent series in which the ratio of one term to the preceding approaches zero. It
follows that we can find a prime number p so large that the equation (17) is
impossible ; hence Hermite's theorem is proved.
8S. Legendre's polynomials. Let us consider the integral
/.
QP«dx,
where P^ (z) is a polynomial of degree n and Q is a polynomial of degree leei
than n, and let us try to determine P«,(x) in such a way that the integral van-
ishes for any polynomial Q. We may consider P„ (z) as the nth derivative of a
polynomial R of degree 2»i, and this polynomial li is not completely determined,
for we may add to it an arbitrary polynomial of degree (n - 1) without changing
its nth derivative. We may therefore set 2^= d^'R/dz"^ where the polynomial /?,
together with its first (n — 1) derivatives, vanishes for z = a. But integrating
by parts we find
Ja dz* L d«-» ^ dx-« ^ d*--U.
and since, by hypothesis,
«(a) = 0, /?'(a) = 0, , «<--»(a)=0,
the expression
'Q(6)f^«-')(6) - Q'(6)/J<'.-t>(6)-f •• ± Q(— ')(6)iJ(6)
must also vanish if the integral is to vanish.
Since the polynomial q of degree n - 1 Is to be arbitrary, the quantitiea
Q(^)i Q'i^)* •• •> 0^""'^^) ^^ themselves arbitrary; hence we must also have
R (6) = 0, R\b) =0, .... iJ<« - »)(6) = 0.
The polynomial R (z) is therefore equal, save for a conctant factor, to the product
(z — o)*(z - 6)" ; and the required polynomial Pmi!'') is completely determined,
•ave for a constant factor, in the form
If the limits a and 6 are — 1 and + 1, respectively, the polynomials P« are
Legendre^s polynomials. Choosing the constant C with Legendre, we will set
174 DEFINITE INTEGRALS [IV, §88
If we alao agree to set Xo = 1, we shall have
^ 3xa-l ^ 6x3 -3x
jro = i, Jri = x, Xt = — - — 1 ^8 = — - — '
In general, X, ta a polynomial of degree n, all the exponents of x being even or
odd with n. Leibniz' formula for the nth derivative of a product of tv^o factors
(S 17) gi^M ^ 0°^ ^® formula
(W) -r,(l) = 1, X.(- 1) = (- 1)».
By the general property established above,
(90) J^ Xn<f>{x)dx = 0,
where i>{z) is any polynomial of degree less than n. In particular, if m and n
are two different integers, we shall always have
(21) f^XmXndx^O.
This formula enables us to establish a very simple recurrent formula between
three successive polynomials X„. Observing that any polynomial of degree n
can be written as a linear function of Xo, Xi, • • . , X„, it is clear that we may set
XX, = CoXn + l + CiXn + CiXn-l + 0^X^-2 +"',
where Co, Ci, Ca, • • • are constants. In order to find Cg, for example, let us
multiply both sides of this equation by Xn-2, and then integrate between the
limita — 1 and + 1. By virtue of (20) and (21), all that remains is
Cs£'[x;_2dx = 0,
and hence Cz = 0. It may be shown in the same manner that C4 = 0, Cs = 0, • • .
The coefficient Ci is zero also, since the product xX„ does not contain x». Finally,
to find Co and Ca we need only equate the coefficients of x" + ^ and then equate
the two sides f or x = L Doing this, we obtain the recurrent formula
(S2) (n + l)X,+i - (2n + l)xXn + nX„_i = 0,
which affords a simple means of calculating the polynomials X„ successively.
The reUtion (22) shows that the sequence of polynomials
(48) Xo, Xi, Xa, •••, Xn
the properties of a Sturm sequence. As x varies continuously from - 1
to 4- 1| the number of changes of sign in this sequence is unaltered except when
through a root of X„ = 0. But the formula (19) show that there are n
of sign in the sequence (23) for x = - 1, and none for x = 1. Hence
the eqaatloD X. = has n real roots between - 1 and + 1, which also readily
follows from Rollers theorem.
IV, §89] IMPROPER AND LIKE INTEGRALS 176
IV. GENERALIZATIONS OF THE IDEA OF AN INTEGRAL
IMPROPER INTEGRALS LINE INTEGRALS •
89. The integrand becomes infinite. Up to the present we have sup-
posed that the integrand remained finite between the limits of inte-
gration. In certain cases, however, the definition may be extended
to functions which become infinite l>etween the limits. Let us first
consider the following particular case : f{x) is continuous for every
value of X which lies between a and 6, and for x = 6, but it becomes
infinite for x — a. We will suppose for definiteness that a < 6.
Then the integral of /(a;) taken between the limits a -|- c and
h (<>0) has a definite value, no matter how small c be taken. If
this integral approaches a limit as < approaches zero, it is usual and
natural to denote that limit by the symbol
X
\x)dx.
If a primitive ai f{x)y say ^(-c), be known, we may write
f
and it is sufficient to examine F(a -\- e) for convergence toward a
limit as e approaches zero. We have, for example,
f Mdx _ M r 1 !_■] .
If fi > 1, the term l/c**"* increases indefinitely as e approaches zero.
But if ^ is less than unity, we may write l/c^-*= e'"**, and it is
clear that -this term approaches zero with e. Hence in this case
the definite integral approaches a limit, and we may write
' Mdx ^ 3/(&~aV-^
{x — ay 1 - ^
If ^ — 1, \v.> liave
X'
and the right-hand side increases indefinitely when c approaches zero.
To sum up, the necessary and stiffirient condition that the given int&'
gral should approach a limit is that fi should be less than unity.
* It is poMible, if desired, to read the next diapter before reading the cloeiDg sw^
tioDS of this chapter.
176 DEFINITE INTEGRALS [IV, §89
The straight line ar = a is an asymptote of the curve whose equa-
tion is
M
if /A is positive. It follows from the above that the area bounded by
the X axis, the fixed line x = i, the curve, and its asymptote, has a
finite value provided that /x < 1.
If a primitive of f(x) is not known, we may compare the given
integral with known integrals. The above integral is usually taken
as a comparison integral, which leads to certain practical rules which
are sufficient in many cases. In the first place, the upper limit b
does not enter into the reasoning, since everything depends upon the
manner in which f{x) becomes infinite for x~ a. We may therefore
replace h by any number whatever between a and h, which amounts
to writing /, , = X*^ , + X * ■'•^ particular, unless f{x) has an infi-
nite number of roots near ic = a, we may suppose that f{x) keeps
the same sign between a and c.
We will first prove the following lemma ;
Let <^(x) be a function which is positive in the interval (a, b),
and suppose that the integral J^ ^<t>(x) dx approaches a limit as c
approaches zero. Then, if \f(x) | < <^(cc) throughout the whole inter-
valf the definite integral f^f(x)dx also approaches a limit.
lff(x) is positive throughout the interval (a, b), the demonstration
is injimediate. For, since f(x) is less than </> («), we have
f(x)dx < I <f>(x)dx.
Moreover J^^^f(x)dx increases as e diminishes, since all of its ele-
ments are positive. But the above inequality shows that it is con-
stantly less than the second integral ; hence it also approaches a
limit If f(x) were always negative between a and b, it would
be necessary merely to change the sign of each element. Finally,
if the function f(x) has an infinite number of roots near x = a, we
may write down the equation
£^A^)dx ^j^^ifix) + I f(x) \]da^ -J' \f(x) I dx.
The Moond integral on the right approaches a limit, since
!/(«)! <^(«). Now the function /(ic) + |/(x)| is either positive
IV. §89] IMPROPKR AND LINE INTEGRALS 177
or zero between a and b, and its value cannot exceed 2 ^(2); hence
the integral
f i/w+i/(»')i]«fa
«/a-f«
also approaches a limit, and the lemma is proved.
It follows from the above that if a function /(x) does not approach
any limit whatever for x = a, but always remains less than a fixed
number, the integral approaches a limit. Thus the inte^^ral
^* sin (1/x) dx has a perfectly definite value.
Practical rule. Suppose that the function f(x) can be written in
the form
f(x)= ^(^) ,
-^"^ ^ (x-a)''
where the function ^(a;) remains finite when x approaches a.
If fjL<l and the function ^(x) remains less in absolute value than
a fixed number A/, the integral approaches a limit. But if fi>l and
the absolute value of \f/(x) is greater than a positive number w, the
integral approac/ies 710 limit.
The first part of the theorem is very easy to prove, for the abso-
lute value of f{x) is less than Af/(x — a)**, and the integral of the
latter function approaches a limit, since fi<l.
In order to prove the second part, let us first observe that ^(ar)
keeps the same sign near x = a^ since its absolute value always
exceeds a positive number m. We shall suppose that ^(x)>0
between a and b. Then we may write
£/<">- >x;.F^
and the second integral increases indefinitely as c decreases.
These rules are sufficient for all cases in which we can find an
exponent fi such that the product (x — aYf{x) approa^'hes, for
X = a, a limit A' different from zero. If fi is less than unity, the
limit b may be taken so near a that the inequality
holds inside the interval (a, 6), where Z. is a positive number graatar
178 DEFINITE INTEGRALS [IV, §89
than I A" |. Hence the integral approaches a limit On the other hand,
if ^ > 1, 6 may be taken so near to a that
inside the interval (a, h), where I is a positive number less than |iir|.
Moreover the function f{x), being continuous, keeps the same sign ;
hence the integral /^"^^ /(a:) c^x increases indefinitely in absolute
value.*
Examples. Let /(a;) = P/Q be a rational function. If a is a root
of order m of the denominator, the product {x - aYf{x) approaches
a limit different from zero for x = a. Since m is at least equal to
unity, it is clear that the integral £^^^/(x)<;a; increases beyond all
limit as c approaches zero. But if we consider the function
where P and R are two polynomials and R{x) is prime to its deriv-
ative, the product (x — ay^^f(x) approaches a limit for a; = a if a
is a root of R(x), and the integral itself approaches a limit. Thus
the integral
dx
f.
i + <
vn^
approaches 7r/2 as c approaches zero.
Again, consider the integral J^^logxdx. The product a;^/^loga;
has the limit zero. Starting with a sufficiently small value of x, we
may therefore write log x < Mx' ^/^, where 3/ is a positive number
chosen at random. Hence the integral approaches a limit.
Everything which has been stated for the lower limit a may be
repeated without modification for the upper limit b. If the function
/ (x) is infinite for a; = 6, we would define the integral j^ /(«) dx to be
the limit of the mtegvdX j^~' f(x)dx as «' approaches zero. lif(x)
is infinite at each limit, we would define J VC^)^^ ^is the limit of
the integral^ ~*f(x)dx as c and t' both approach zero independ-
ently of each other. Let c be any number between a and b. Then
we may write
*The flrat part of the proposition may also be stated as follows: the integral has
a limit If an exponent m can \w fiuind (0 < m < 1) such that the product (x — a)'*/(x)
approaclMt a limit ^ aa x u^tpruaches a, — the case where A = uot being excluded.
IV,}90] IMPROPER AND LINE INTEGRALS 179
r"f{x)dx = r f(x)dx + r"f(?:)dx,
and each of the integrals on the right should approach a limit in
this case.
Finally, if f(x) becomes infinite for a value e between a and b,
we would define the integral j^f{x)dx as the sum of the limits of
the two integrals j^~*f{x)dx, j^ ^f{x)dxy and we would proceed
in a similar manner if any number of discontinuities whatever lay
between a and b.
It should be noted that the fundamental formula (6), which was
established under the assumption that f{x) was continuous between
a and h^ still holds when f(x) becomes infinite between these limite,
provided that the primitive function F(x) remains continuous. For
the sake of definiteness let us suppose that the function /(ar) becomes
infinite for just one value c between a and b. Then we have
f(x) dx = \im I f{x) (& + lim / f{x) dx ;
and if F(x) is a primitive of /(x), this may be written as follows :
Xf{x)dx = lim F(c -t!)- F(a) -f F{b) - lim F{c -|- c).
Since the function F(x) is supposed continuous for ar = r, F(c + e)
and F(c — «') have the same limit F(c), and the formula again
becomes
f(x)dx = F(b)-F(,a).
£
The following example is illustrative :
i
£=[:-']:;-.
If the primitive function F(x) itself becomes infinite l^etween a and
b, the formula ceases to hold, for the integral on the left has as yet
no meaning in that case.
The formulae for change of variable and for integration by parts
may be extended to the new kinds of integrals in a similar manner
by considering them as the limits of ordinary integrals.
90. Infinite limits of integration. Let/(a;) be a function of x which
is continuous for all values of x greater than a certain number a.
Then the integral /'/(x)<ir, where / > a, has a definite value, no
X80 DEFINITE INTEGRALS [IV, §90
matter how large I be taken. If this integral approaches a limit
as / increases indefinitely, that limit is represented by the symbol
X
f(x)dx.
If a primitive of f{x) be known, it is easy to decide whether the
integral approaches a limit. For instance, in the example
^ dx
i
„ = arc tan I
^ + ^'
the right-hand side approaches 7r/2 as I increases indefinitely, and
this is expressed by writing the equation
dx IT
X
l-{-x^ 2
Likewise, if a is positive and /x — 1 is different from zero, we have
r'kdx^ k /J 1_\
If fi is greater than unity, the right-hand side approaches a limit as
I increases indefinitely, and we may write
r^'^ kdx ^
Ja X'^
k
On the other hand, if fi is less than one, the integral increases indefi-
nitely with l. The same is true for /x = 1, for the integral then
results in a logarithm.
When no primitive of f(x) is known, we again proceed by com-
parison, noting that the lower limit a may be taken as large as we
please. Our work will be based upon the following lemma :
Let ^(x) be a function which is positive for x'> a, and suppose that
the integral JJ <f> (x) dx approaches a limit. Then the integral J ^f(x) dx
also approaches a limit provided that \f(x) \<<f»(x) for all values of
X greater than a.
The proof of this proposition is exactly similar to that given above.
If the function f(x) can be put into the form
wli«re the function ^{x) remains finite when x is infinite, the follow-
ing theorems can be demonstrated, but we shall merely state them
IV. §91] IMPROPER AND LINE INTEGRALS 181
If the absolute ralue o/\f/(^x) I'.s /'.s.s f/f/n >i fix*''l numhtr M ,i ml
fi is (jreater than unity, the integral njipronrht^ a limit.
If the absolute value of ^ (x) is greater than a potitive number m
arul /i is less than or equal to unity^ the integral approaehee no limit.
For instance, the integral
X
COB ax ,
ax
/o 1+x^
approaches a limit, for the integrand may be written
cos ax __ \ cos ax
and the coefficient of 1/x* is less than unity in absolute value.
The above rule is sufficient whenever we can find a positive num-
ber /i for which the product x^f(x) approaches a limit different from
zero as x becomes infinite. The integral approaches a limit if /i is
greater than unity, but it approaches no limit if fi is less than or
equal to unity.*
For example, the necessary and sufficient condition that the inte-
gral of a rational fraction approach a limit when the upper limit
increases indefinitely is that the degree of the denominator should
exceed that of the numerator by at least two units. Finally, if we
take
' V/e(x)
where P and R are two polynomials of degree p and r, respectively,
the product a;'"/*~»'/(x) approaches a limit different from zero when
X becomes infinite. The necessary and sufficient condition that the
integral approach a limit is that p be less than r/2 —1.
91. The rules stated above are not aiw«iy8 sufiii'ient for deiermin-
ing whether or not an integral approaches a limit. In the example
/(j-) = (sin x)/x, for instance, tlie product x^f(x) approaches zero if
fi is less than one, and can take on values greater than any given
number if /i is greater than one. If /a = 1, it oscillates between + 1
and — 1. None of the above rules apply, but the integral does ap-
proach a limit Let us consider the slightly more general integral
* The integral also approaches a limit if the product a^/(x) (where m> 1) approadiag
zero ai) x becomes intiuite.
182 DEFINITE INTEGRALS [IV, §91
-i:
■" dXf a>0.
The integrand changes sign for x = kw. We are therefore led to
study the alternating series
(24) ao-a, + a^-a, + '-' + (-iya, + "',
where the notation used is the following :
^^ sm X .
-«^ dx
J- /^Cn + l)*
nv
Substituting y + wtt for x, the general term a„ may be written
»"=X'^"
It is evident that the integrand decreases as n increases, and hence
«•+!<"••• Moreover the general term a„ is less than J^ (l/7i7r)c?y,
that is, than \/n. Hence the above series is convergent, since the
absolute values of the terms decrease as we proceed in the series,
and the general term approaches zero. If the upper limit / lies
between nir and (n + 1) 77 , we shall have
X'
X
where S^ denotes the sum of the first n terms of the series (24). As
/ increases indefinitely, n does the same, a„ approaches zero, and the
integral approaches the sum S of the series (24).
In a similar manner it may be shown that the integrals
I sXnx^dx, I QO^x^dXy
Jo
which occur in the theory of diffraction, each have finite values.
The curve y = sin x^, for example, has the undulating form of a sine
curve, but the undula tions becom e sharper and sharper as we go out»
since the difference v(n-i-l)7r — y/nir of two consecutive roots of
sin x^ approaches zero as n increases indefinitely.
Bimarlr. Thli last example gives rlae to an interesting remark. As x increases
todsflBltoly sin x« oscillates between - 1 and +1. Hence an integral may
approach a limit uvea if tlm integrand does not approach zero, that is, even if
IV, i'-'J IMPUorKK AND LIXE INTEGRALS 188
the X axis Is not an a«y rnptote to tlie curve y — f{x). The following Is an example
of the same kind in which the function /(z) does not change sign. The function
m
1 + «• sin«x
remains positive when x is positive, and it does not approach zero, since
f{kn) - kit. In order to show that the integral approaches a limit, let us con-
sider, as above, the series
oo + ai + ••• + a, + --,
where
-L
xax
+ x«ain«x
As X varies from twr to (n + 1) «•, x« is constantly greater than n««*, and we may
write
a^<(n-|-l)«' \
%Jhw
(-^»' dx
1 + n«»r«8ln2x
A primitive function of the new integrand is
arctan(Vl4- n»7r< tanx),
and as x varies from nn to (n + 1) >r, tanx becomes infinite just once, p— nng
from + 00 to — 00. Hence the new integral is equal (§ 77) to ir/Vl + n^««, and
we have
^^ (n + l)>r« ^ (n-fl)
It follows that the series Zom is convergent, and hence the integral fj/{x)dz
approaches a limit.
On the other hand, it is evident that the integral cannot approach any limit
if /(x) approaches a limit h different from zero when x becomes infinite. For
beyond a certain value of x, /(x) will be greater than | h/2 \ in absolute value
and will not change sign.
The preceding developments bear a close analogy to the treatment of infinite
series. The intimate connection which exists between these two theories is
brouj^ht out by a theorem of Cauchy's which will be considered later (Chapter
VIII). We shall then also find new criteria which will enable us to determine
whether or not an integral approaches a limit in more general cases than those
treated above.
98. The function r(s). The definite integral .
(26) r(a)= r^*x«->c-»dx
has a determinate value provided that a is positive.
Fur, let us consider the two integmls
r x--»e-'<ix, rx«->e-«cfc.
184 DEFIXTTK INTEGRALS [IV, §93
when c is a Tery small positive number and I is a very large positive number.
The aeoond integral always approaches a limit, for past a sufficiently large value
of X we have x^-U-' < l/x*, that is, e^>x^-^K As for the first integral, the
product x> - "/(3K) approaches the limit 1 as x approaches zero, and the necessary
and sufficient condition that the integral approach a limit is that 1 - a be less
unity, that is, that a be positive. Let us suppose this condition satisfied.
the sum of these two limits is the function T{a), which is also called Euler^s
inteffral of the second kind. This function r(a) becomes infinite as a approaches
»ero, it is positive when a is positive, and it becomes infinite with a. It has
a minimum for o = 1.4616321 •• •, and the corresponding value of T{a) is
0.8866082. •.
Let us suppose that a> 1, and integrate by parts, considering e-*dx as the
differential of - e-^. This gives
r(o) = -[x«-ie-']^* + (a-l)J *a;«-2e-^(te,
hut the product x^-^e-' vanishes at both limits, since a > 1, and there remains
only the formula
(26) r(a) = (a-l)r(a-l).
The repeated application of this formula reduces the calculation of r(a) to
the case in which the argument a lies between and 1. Moreover it is easy to
determine the value of T{a) when a is an integer. For, in the first place,
r(l) = j^''"e-dx = -[e-],+ * =1,
and the foregoing formula therefore gives, for a = 2, 3, • • , n • • • ,
r(2) = r(i) = 1, r(3) = 2r(2) = i . 2 ;
and, in general, if n is a positive integer,
(27) r(n) = 1.2.3...(n-l) = (n-l)!.
93. Line integrals. Let ^J5 be an arc of a continuous plane curve,
and let P(x, y) be a continuous function of the two variables x and
y along AB, where x and y denote the coordinates of a point of AB
with respect to a set of axes in its plane. On the arc AB let us
take a certain number of points of division Wi, Wa, • • •, w-,, • • •, whose
ootedinates are (a^x, y^), (ajj, y,), • • -, (a;,, y^), • • -, and then upon each
of the arcs m^^jm^ let us choose another point n,- (^,, i;,) at random.
Finally, let us consider the sum
(28) S ^^^'' "^'^ (^» ~ *) + ^(^» ' ''«) (^« -^i)+"
extended oyer all these partial intervals. When the number of points
of diTision is increased indefinitely in such a way that each of the
diiferenoes «< — x, _ , approaclies zero, the above sum approaches a
IMPROPER AVn LIXK IXTEORALS
185
iuiiil which i.s oalltMl tl
arc^ Ali, and which is r.
u ut J\Xf y) extemied over the
the Bymbol
/
Jab
P{x, y)dx.
In order to establish the existence of this limit, let us first sup-
pose tliat a lino parallel to the y axis cannot meet the arc AH xn
more tlian one point. Let a and b be the abscissae of the points A
and Hy respectively, and let y = ^(aj) be the equation of Uie curve ^4 /J.
Then ^(j*) is a continuous function of x in the interval (a, 6), by
hypothesis, and if we replace y by <^(j-) in the function /'(ar, y), the
resulting function ♦(x) = P\xj ^(^)] Js also continuous. Hence we
have
and the preceding sum may therefore be written in the form
♦(^i) (a-i - a) +*(f,) (x, - xO -f •• • + <I>(«(ar. - x,.,) + ....
It follows that this sum approaches as its limit the ordinary definite
integral
%J a %J a
a!i<] \v»' li !v.' filially the rnrmnl.i
f I'(x,i,)Ux=f Plx,4,(x)-idx.
Jab Ja
If a line parallel to the y axis can meet the arc AB in more than
one point, Sve should divide the arc
into several portions, each of which
is met in but one point by any line
parallel to the y axis. If the given
arc is of the form ACDIS (Fig. 14),
for instance, where C and D are
points at which the abscissa has an
extremum, each of the arcs ACj CDy
I>B satisfies the above condition, and
we may write
r P(x, y)dx = r P(x, y)dx + r P(x, y)djr-^ f P(x, y)dx,
Javhb Jac Jcd Jdb
But it should be noticed that in the calculation of the three integrals
y
^D
a
!\
C
A^
i
I !
X
Fio.
14
Ig5 DEFINITE INTEGRALS [IV, §93
on the right-hand side the variable y in the function P{x, y)
must be replaced by three different functions of the variable Xy
respectively.
Curvilinear integrals of the form J^^ Q,(x, y) dy may be defined
in a similar manner. It is clear that these integrals reduce at once
to ordinary definite integrals, but their usefulness justifies their
introduction. We may also remark that the arc AB may be com-
posed of portions of different curves, such as straight lines, arcs of
circles, and so on.
A case which occurs frequently in practice is that in which the
coordinates of a point of the curve AB are given as functions of a
variable parameter
where <^(^) and \l/(t), together with their derivatives <^'(^) and \f/\t),
are continuous functions of t. We shall suppose that as t varies
from a to p the point (x, y) describes the arc AB without changing
the sense of its motion. Let the interval (a, p) be divided into a
certain nimiber of subintervals, and let ^,-_ i and t^ be two consecu-
tive values of ^ to which correspond, upon the arc AB, two points
m,._, and m,- whose coordinates are (ic,_i, 2/,_i) and (a,-, y^), respec-
tively. Then we have
where d,- lies between ^,._i and ^,. To this value Oi there corresponds
a point (^,-, i;,) of the arc m,_im,; hence we may write
SP(^o 1,,) (a:, - x,_,) = SP[<^(eO. «AW)] <^'(^i) (ti - *i-i)>
or, passing to the limit,
/ Pix, y)dx= f P[<f>(t), ,/r(0] <t>\t)dt.
Jab Ja
An analogous formula for JQdy may be obtained in a similar manner.
Adding the two, we find the formula
(29) I Pdx -\-Qdy= f lP<t>'(t) + Qxif'mdt,
Jab Ja
which 18 the formula for change of variable in line integrals. Of
course, if the arc AB ia composed of several portions of different
curves, the functions 4>(t) and \l/{t) will not have the same form
along the whole of A B, and the formula should be applied in that
to each portion separately.
IV. $H4] IMPROPFU AVT) ITVK TVTKnnAT.^i 187
94. Area of a closed curv< :y deiiueU the area of a
portion of the ])lunu Ix>uii(h / /i, & straight line which
does not out that arc, and the two perpendiculars AA^t BBq let fall
from the points A and B upon the straight line (§§ G5, 78, Fig. 9).
Let us now consider a continuous closed curve of any shape, fay
which we shall understand the locus described by a point M whose
coordinates are continuous functions x =/(/), y = 4*{0 ^^ * param-
eter t which assume the same values for two values to and 7* of
the parameter t. The functions /(t) and ^(/) may have several
distinct forms between the limits t^ and 7'; such will be the case,
for instance, if the closed contour C be composed of portions of
several distinct curves. Let A/© , 3/i , 3/, , • • • , 3/< _ , , i/< , • • • , A/, „ , , iV©
(Iriiote points upon the curve C corresponding, respectively, to the
vahu's /o» Ui fij '•'■> ^i-\i ^ii ••» ^w-u ^ of the parameter, which
increase from ^o to 7\ Connecting these points in order by straight
lines, we obtain a polygon inscribed in the curve. The limit
approatrhed by the aiea of this polygon, as the number of sides is
indefinitely increased in such a way that each of them approaches
zero, is called the area of the closed curve (\* This definition is
seen to agree with that given in the particular case treated above.
For if the polygon AqAQ^Q^-- BBqAq (Fig. 9) be broken up into
small trapezoids by lines parallel to .-1.40, the area of one of these
trapezoids is (ar^ - x,_ ,) [/(x,) -f /(x,_ ,)]/2, or (x^ - x,_,)/(^<),
\shere ^, lies between x<_i and a:^. Hence the area of the whole
polygon, in this special case, approaches the definite integral
//(x)rfx.
Let us now consider a closed curve C which is cut in at most two
points by' any line parallel to a certain fixed direction. Let us
choose as the axis of y a line parallel to this direction, and as the
axis of a; a line perpendicular to it, in such a way that the entire
curve C lies in the quadrant xOy (Fig. 16).
The points of the contour C project into a segment ah of the axis
Ox, and any line parallel to the axis of y meets the contour C in at
most two points, t/i, and m,. Let yx = ^j(x) and y, = ^s(x) be the
equations of the two arcs AniiB and Am^By respectively, and let
us suppose for simplicity that the points A and B of the curve C
which project into a and b are taken as two of the vertices of the
* It Is Bappo0«d, of course, that the curve under oonsideraUon has no double point,
and that the Hidtv of the polygon have been chosen so snudl that the polygon itself
has no double point.
188
DEFINITE INTEGRALS
[IV. §94
polygon. The area of the inscribed polygon is equal to the differ-
ence between the areas of the two polygons formed by the lines Aa,
abf bB with the broken lines inscribed in the two arcs Am^B and
AntxBy respectively. Passing to the limit, it is clear that the area
of the curve C is equal to the difference between the two areas
bounded by the contours AmtBbaA and Am^BbaA, respectively, that
is, to the difference between
the corresponding definite in-
tegrals
' il/2(x)dx — / il/i(x)dx.
a %J a
These two integrals represent
the curvilinear integral Jyt^a;
taken first along Am^B and
then along AmiB. If we
agree to say that the contour
C is described in the positive
sense when an observer standing upon the plane and walking around
the curve in that sense has the enclosed area constantly on his left
hand (the axes being taken as usual, as in the figure), then the above
result may be expressed as follows : the area 12 enclosed by the
contour C is given by the formula
Fia. 16
(30)
O = — I ydxy
J(C)
where the line integral is to be taken along the closed contour C in
the positive sense. Since this integral is unaltered when the origin
is moved in any way, the axes remaining parallel to their original
positions, this same formula holds whatever be
the position of the contour C with respect to
the coordinate axes.
Let us now consider a contour C of any form
whatever. We shall suppose that it is possible
to draw a finite number of lines connecting
pain of points on C in such a way that the
retoltiDg Bubcontours are each met in at most
two points by any line parallel to the y axis.
Such is the case for the region bounded by the
contour C in Fig. 1 6, which we may divide into three subregions
bounded by the contours amba, abndcqa, cdpc, by means of the
Fio. 16
iv,§ufl] IMPKOPKK AND LINE INTEGRALS 189
traDBversaU ab and cd. Applying the preceding formula to each
of these subregions and adding the results thus obtained, the line
integrals which arise from the auxiliary lines ab and ed cancel each
otlier, and tlie area bounded by the closed curve C is still given by
the line integral — jy djt taken along the contour C in the positive
sense.
Similarly, it may be shown that this same area is given by the
formula
(31) Q=fxdyi
and finally, combining these two formulae, we have
(32) " = 9 / xdy-ydxy
where the integrals are always taken in the positive sense. This
last formula is evidently independent of the choice of axes.
If, for instance, an ellipse be given in the form
ar = acos^, y = 6sin^,
its area is
1 r"
O = - I a6(cos*^ 4- sin*<) dt = irab.
95. Area of a curve in polar coordinates. Let us try to find the
area enclosed by the contour 0AM HO (Fig. 17), which is composed
of the two straight lines 0.1, OBy and the arc AMB^ which is
met in at most one
point by any radius
vector. Let us take
O as the pole and a
straight line Ox as /
the initial line, and i
let p = /(w) be the V
equation of the arc
A MB.
Inscribing a polygon in the arc AMB^ with A and B as two of
the vertices, the area to be evaluated is the limit of the sum of such
triangles as OMM\ But the area of the triangle OMM^ ia
Flo. 17
^ p(/) H- Ap) sin A« = Awf^ 4-<j,
190 DEFINITE INTEGRALS [IV, §95
where < approaches zero with Aw. It is easy to show that all the
quantities analogous to c are less than any preassigned number rj
provided that the angles A(o are taken sufficiently small, and that
we may therefore neglect the term cAw in evaluating the limit.
Henoe the area sought is the limit of the sum 2/)* Aw/2, that is, it
is equal to the definite integral
2X
where <i>i and ta^ are the angles which the straight lines OA and OB
make with the line Ox.
An area bounded by a contour of any form is the algebraic sum
of a certain number of areas bounded by curves like the above. If
we wish to find the area of a closed contour surrounding the point
O, which is cut in at most two points by any line through 0, for
example, we need only let <u vary from to 27r. The area of a con-
vex closed contour not surrounding (Fig. 17) is equal to the dif-
ference of the two sectors 0AM BO and OANBO, each of which may
be calculated by the preceding method. In any case the area is
represented by the line integral
U'
taken over the curve C in the positive sense. This formula does
not differ essentially from the previous one. For if we pass from
rectangular to polar coordinates we have
X = p cos o>, y = p sin CD,
dx = COS to dp — p sin to du), dy = sin <adp -^ p cos o> doty
X di/ — 7/ dx = p^do).
Finally, let us consider an arc A MB whose equation in oblique
coordinates is y =f(x). In order to find the area bounded by this
arc AM By the x axis, and the two lines AA^, BBq, which are parallel
to the y axis, let us imagine a polygon inscribed in the arc A MB, and
Icfc ua break up the area of this polygon into small trapezoids by
lioet parallel to the y axis. The area of one of these trapezoids is
^"-'K'-^'"\ ^,-^...)sine,
IV, §96] IMPROPER AND LINE INTEGRALS 191
which may be written in the form (x^_^— Xt)/(fi)A\n$, where ^
lies in the interval (x^.d x^). Hence the area in question is equal
to the definite integral
8in^ / /{x)dXf
where Xq and A' denote the abscisssB
of the points A and Bj respectively.
It may be shown as in the similar
case above that the area bounded by
any closed contour C whatever is given
by the formula
-y I xdy-ydx.
Note. Given a closed curve C (Pig. 16), let us draw at any point
M the portion of the normal which extends toward the exterior,
and let a, /3 be the angles which tliis direction makes with the axes
of X and y, respectively, counted from to tt. Along the arc A mj B
the angle /3 is obtuse and cte = — dscosp. Hence we may write
Ji/dx = — j y cos fids.
{Am^B) J
Along BytiiA the angle fi is acute, but dx is negative along Bm^A
in the line integral. If we agree to consider ds always as positive,
we shall still have dx = — ds cos fi. Hence the area of the closed
curve may be represented by the integral
/
ycospdSf
where the angle fi is defined as above, and where d* is essentially
positive. This formula is applicable, as in the previous case, to a
contour of any form whatever, and it is also obvious that the
area is given by the formula
/
r cos It ds.
These siatements are absolutely independent of the choice of axea.
•6. ValM of the intesral /xdy - ydx. Ii Is nmlural to inquire what will
be represented by the integral fzdy — ydz, taken nv«»r ^nv ..irvn whatever,
doeed or oncloeed.
192 DEFINITE INTEGRALS [IV, §97
Let us consider, for example, the two closed curves OAOBO and
ApBqCrAtBtCuA (Fig. 19) which have one and three double points, respec-
tively. It is clear that we may replace either of these curves by a combination
of two closed curves without double points. Thus the closed contour OAOBO
is equivalent to a combination of the
two contours OAO and OBO. The
integral taken over the whole contour
is equal to the area of the portion
OAO less the area of the portion
OBO. Likewise, the other contour
may be replaced by the two closed
curves ApBqCrA and AsBtCuA, and
the integral taken over the whole con-
tour is equal to the sum of the areas of ApBsA, BtCqB, and ArCuA , plus twice
the area of the portion AsBqCuA. This reasoning is, moreover, general. Any
closed contour with any number of double points determines a certain number
of partial areas <ri, <r2, • • •, <rp, of each of which it forms all the boundaries.
The integral taken over the whole contour is equal to a sum of the form
miffi + m^ff^ 4- • • • + mpo-p,
where mi, m^, • • •, wip are positive or negative integers which may be found by
the following rule : Given two adjacent areas <r, <r', separated by an arc ab of the
contour C, imagine an observer walking on the plane along the contour in the sense
determined by the arrows ; then the co^cient of the area at his left is one greater
than that of the area at his right. Giving the area outside the contour the coeflB-
cient zero, the coefficients of all the other portions may be determined successively.
If the given arc AB'\& not closed, we may transform it into a closed curve by
joining its extremities to the origin, and the preceding formula is applicable to
this new region, for the integral fxdy — ydx taken over the radii vectores OA
and OB evidently vanishes.
V. FUNCTIONS DEFLN^ED BY DEFINITE INTEGRALS
97. Differentiation under the integral sign. We frequently have to
deal with integrals in which the function to be integrated depends
not only upon the variable of integration but also upon one or more
other variables which we consider as parameters. Let f(x, a) be a
continuous function of the two variables x and a when x varies from
r^U) X and a varies between certain limits ctq and ai. We proceed
to study the function of the variable a which is defined by the
definite integral
where a is supposed to have a definite value between «„ and aj, and
where the limits x^ and X are independent of a.
Iv.iVT] FUNCTIOIJS DEFINED BY INTEGRALS 198
We have then
(33) F(a + Aa) - F(a) = f [/(x, a + Aar) -/(x, a)] dr.
Since the function /[x, a) is continuous, this integrand may be made
less than any preassigned number c by taking Aa sufficiently small.
Hence the increment ^F{a) will be less than €\X — x^\ in absolute
value, which shows that the function F(a) is continuous.
If the function f(Xf a) has a derivative with respect to a, let us
write
f(x, a -f Aa) ^/(x, a) = Aa [/. (x, a) + t] ,
where c approaches zero with A<r. Dividing both sides of (33) by
Aa, we find
F(a 4- Aa) - F(a) f\ . ^ C' .
and if 17 be the upper limit of the absolute values of c, the absolute
value of the last integral will l)e less than ri\X — Xq\. Passing to
the limit, we obtain the formula
In order to render the above reasoning perfectly rigorous we most
show that it is possible to choose Aa so small that the quantit}' c
will be less than any preassigned number rj for all values of r l^etween
the given Jimits Xq and A'. This condition will certainly be satisfied
if the derivative /«(x, a) itself is continuous. For we have from
the law^of the mean
/(x, a 4- Aa) -/(x, a) = Aa/, (x, a -f tf Aa), < tf < 1,
and hence
€=M^,^'he^a)-/,(x,a),
If the function/, is continuous, this difference c will be less than if
for any values of x and a, provided that i Aa| is less than a properly
chosen positive number h (see Chapter VI, $ 120).
I^t us now suppose that the limits A' and x, are themselves func-
tions of a. If A A' and Aj-q denote the increments which correspond
to an increment Aa, we shall have
194 DEFINITE INTEGRALS tIV,§97
Jan,
/(a;,a + Aar)<i:aj
I /(x, a4-AQ:)rfa;;
or, applying the first law of the mean for integrals to each of the
last two integrals and dividing by Aa,
F(a-\-Aa)-F(a) ^ C "" f(x, a + Aa) - f(x, a) ^^
Aa J^^ Aa
AX
-\- — f(X -^ e AX, a + Aa)
Aor
-■^f(x,-^e'Ax,,a-\-Aa).
As Aa approaches zero the first of these integrals approaches the
limit found above, and passing to the limit we find the formula
which is the general formula for differentiation under the integral
sign.
Since a line integral may always be reduced to a sum of ordinary
definite integrals, it is evident that the preceding formula may be
extended to line integrals. Let us consider, for instance, the line
integral
F(a) = I P{xy y,a)dx-ir Q(x, ij, a)dy
Jab
taken over a curve A B which is independent of a. It is evident that
we shall have
Jai
Pa(Xy y, a)dx + Q„{x, y, a)dy,
where the integral is to be extended over the same curve. On the
other hand, the reasoning presupposes that the limits are finite and
that the function to be integrated does not become infinite between
the limits of integration. We shall take up later (Chapter VIII,
f 175) the cases in which these conditions are not satisfied.
IV, §98] FFVCTIONS PFFTVKD BY INTF.ORAI^i 196
The foiiiiuhi ^^35) is frequuutly i
integrals by reducing them to oth« i i u
lated. Thus, if a is poeitiye, we have
/■
€1x1 X
-J- — = -p arc tan -7= >
whence, applying the formula (34) n — 1 times, we find
(-l).-.1.2...(»-l)jr'^. = £^(^are,a„^).
9S. Examplet of discontinuity. If the conditions imposed are not satisfied for
all valaes between the liinitH of integration, it may happen that the definite Inte-
gral defines a discontinuous function of the parameter. Let us consider, for
example, the definite integral
»/ V r sinadx
F{a)
■i>-
2x cos a + 2^
This integral always has a finite value, for the roots of the denominator are
imaginary except when <r = krt, in which case it is evident that f\a) — 0. Sup-
posing that sin a ;e and making the substitution z = cos a -^ t sin a, the indefi-
nite integral becomes
- — -= I ; — - =arctan<.
1 - 2xcosar + x« J l + (^
Hence the dt finite integral F{a) has the value
^ /I — cosaX ^ /-I — cosa\
arc tan ( — ; ) — arctani ) «
\ sina / \ sina /
where the angles are to be taken between - x/2 and x/2. But
1 — cosa — 1 — cosa
sin or sin a
= -1,
and hence the difference of these angles is ± n/2. In order to determine the
sign uniquely we need only notice that the sign of the integral is the same as
that of sin a. Hence F(a) = db n/2 according as sin a is positive or negative.
It follows that the function F{a) is discontinuous for all values of (i: of the form
At. This result does not contradict the above reasoning in the least, however.
For when x varies from — 1 to -f 1 and a varies from — c to + t, for example,
the function under the integral sign assumes an indeterminate form for the sets
of values nr = 0, x = - 1 and a = 0, x = + 1 which belong to the region in qnee-
tion for any value of 9.
It would be easy to give numerous examples of this nature. A|;ain. consider
the integral
* * sin mx
I
dx.
196 DEFINITE INTEGRALS [IV, §99
the snbetltution mx = y, we find
X"*" * sin mx , , r
— -dy,
where the aign to be taken is the sign of m, since the limits of the transformed
integral are the same as those of the given integral if m is positive, but should
be interehanged if m is negative. We have seen that the integral in the second
member is a positive number JV (§ 91 ). Hence the given integral is equal to ± ^
according as m is positive or negative. If m = 0, the value of the integral is
rero. It is evident that the integral is discontinuous for m = 0.
VI. APPROXIMATE EVALUATION OF DEFINITE INTEGRALS
99. Introduction. When no primitive of f(x) is known we may-
resort to certain methods for finding an approximate value of the
definite integral Cf(x) dx. The theorem of the mean for integrals
furnishes two limits between which the value of the integral must
lie, and by a similar process we may obtain an infinite number of
others. Let us suppose that <f>{x) <f(x) < i{/(x) for all values of x
between a and b (a< b). Then we shall also have
<l>(x)dx< I f(x)dx< I \l/(x)dx.
U a %J a
If the functions ^{x) and ^{x) are the derivatives of two known
functions, this formula gives two limits between which the value of
the integral must lie. Let us consider, for example, the integral
=x-
dx
Now Vl-x* = Vl-a;« Vl + x^, and the factor VlT^ lies
between 1 and V2 for all values of x between zero and unity.
Hence the given integral lies between the two integrals
dx
Jo Vl-a;«' V2 Jo •
that Ib, between 7r/2 and 7r/(2V2). Two even closer limits may-
be found by noticing that (l-\-x^)-^^^ is greater than 1 — a72,
which results from the expansion of (1 -f u)-^^^ by means of Taylor's
series with a remainder carried to two terms. Hence the integral
/ is greater than the expression
r dx 1 r
Jo vr=^ 2jo
x^dx
IV. §wj APPROXIMATE EVALUATION' 197
Thu stMoiid of these integrals has the value ir/4 (§ 105) ; hence /
lies between 7r/2 and 3 tt/S.
It is evident that the preceding methods merely lead to a rough
idea of the exact value of the integral. In order to obtain closer
approximations we may break up the interval (a, b) into smaller
subintervals, to each of which the theorem of the mean for inte-
grals may be applied. For deiiniteness let us suppose that the
function f(x) constantly increases as x increases from a to b. Let
us divide the interval (a, b) into n equal parts (b — a = nh). Then,
by the very definition of an integral, j^f{x)dx lies between the
two sums
' = Aj/(a) +/(«4-/0 4- ••+/[«+ (n-l)A]{,
S = h\f(a + h) -\-f(a + 2A) + • +/(« -f nh)\,
I f we take (S -h *)/2 as an approximate value of the integral, the
error cannot exceed |5-«|/2 = |[(6 - a)/2 w][/(*)-/l(a)]|. The
value of (.s -f «)/2 may be written in the form
jt ( /['')+ /(« + *) , /(« + /0+/^» + 2A) ^
(2 ^
"^ 2 i'
Observing that j/(a -f »A) +/[a + (» 4-l)A]jA/2 is the area of
the trapezoid whose height is h and whose bases SLref(a + ih) and
/{a + ih -H A), we may say that the whole method amounts to
replacing the area under tlie curve y = f(x) between two neighbor-
ing ordinates by the area of the trapezoid whose bases are the two
ordinaties. This method is quite practical when a high degree of
approximation is not necessary.
Let us consider, for example, the integral
i:
dx
Taking 7) = 4, we find as the approximate value of the integral
and the error is less than 1/16 = .0625.* This gives an approxi-
mate value of IT which is correct to one decimal place, — 8.1311 • • •.
• Found from Uie formula \a — »\/% In fact, Um error is abont .00880, the exaot
value being it/\.—1iUi»:
198 DEFINITE INTEGRALS [IV, §100
If the function f{x) does not increase (or decrease) constantly as
X increases from a to i, we may break up the interval into sub-
intervals for each of which that condition is satisfied.
100. Interpolation. Another method of obtaining an approximate
value of the integral f^f(x) dx is the following. Let us determine
a parabolic curve of order n,
y = 4>(x) = tto + ai* H h a„a5",
which passes through {n -f- 1) points Bq^ B^ •••, B„ of the curve
y —f{x) between the two points whose abscissae are a and b.
These points having been chosen in any manner, an approximate
value of the given integral is furnished by the integral f^<f>(x)dx,
which is easily calculated.
Let (xof yo), («n yi)i • • > (^«> ^n) be the coordinates of the (n+l)
points Boy Biy •••, B„. The polynomial <l>(x) is determined by
Lagrange's interpolation formula in the form
^(x) = yoXo+yi^i+-- + y,X, + ... + y„Z„,
where the coefficient of y,- is a polynomial of degree n,
* (x, - Xo) • • • i^i - Xi_{) (X, -X,^,)-'- (Xi - x„) '
which vanishes for the given values cco, aji, • •, a;„, except for a: = a;,.,
and which is equal to unity when a; = «,. Hence we have
' <^(x)rfx = y Vil X,dx.
a f^Q Ja
The numbers x^ are of the form
«o = a + ^o(* - a), aji = a H- B^{b - a), . . ., «„ = a + BJh - a),
where < do < ^i < • • • < d„ 5 1. Setting x = a ■\- {h - a)t, the ap-
proximate value of the given integral takes the form
(36) (^-a)(^oyo + /^iyi+-+A',y.),
where K^ is given by the formula
/r,= r (^-^o)''(^-<?.-i)(^-g...,)..(^-gj
Jo (^i-^o)--(^.-^,-OW-^.>i)-(^.-^„)
If we divide the main interval (a, h) into subintervals whose
ratios are the same constants for any given function /(a;) whatever,
the numbers do, ^i, • ■ , d,, and hence also the numbers A'.., are inde-
pt'udent of /(x). Having calculated these coefficients once for all,
I\,ii.Mj APPROXIMATE KVALUATION 199
it only remains to replace j/ot !/if "t Vm^ ^^^i' retpeotiTe yalues
in the formula (36).
If the curve f(x) whoee area is to be evaluated is given graph-
ically, it is convenient to divide the interval (a, b) into equal part*,
and it is only necessary to measure certain equidistant ordinates of
this curve. Thus, dividing it into halves, we should take $^ = 0,
di = 1/2, d, = 1, which gives the following formula for the approxi-
mate value of the integral :
b-a
/=-g— Cyo + 4y, 4-yi).
Likewise, for n = 3 we find the formula
/ = ^-^ (yo + 3y» + 3y, -f y.),
and for n = 4
/ = -^'^ (7yo + 32y, + 12y, + 32y. + Ty,).
The preceding method is due to Cotes. The following method,
due to Simpson, is slightly different. Let the interval (a, b) be
divided into 2n equal parts, and let yo» yn y«» ••» y*. be the ordi-
nates of the corresponding points of division. Applying Cotes*
formula to the area which lies between two ordinates whose indices
are consecutive even numbers, such as y© and y,, y, and t/t, etc., we
find an approximate value of the given area in the form
/ = -^"^[(yo -f 4yi + y,) + (y, 4- 4y, + yO -^ • • •
+ (y«.-f + 4y»,_, + y^)],
whence, upon simplification, we find Simpson's formula:
b- a
^ = -^ [yo 4- yx., T - v.y3 -r y« -r • • • 4- t/u^t)
4-4.',/. -^,/. 4-...4.y^.,)].
101. Gauss' method, in dauss m* iii<>d other values are assigned
the quantitie.s $^. The arguiiuiit is :is follows: Suppose that we
ran find polynomials of increasing <: iiich differ less and less
from the given integrand /(x) in luu interval (a, b). Suppose,
for instance, that we can write
/(x) = ao + <»i* + a,a;« + • • • -h «tie.i«*'"* + Ru(')^
where the remainder R^i^) is ^^^^ ^^^^^^ ^ fixti! number < for all
200 DEFINITE INTEGRALS [IV. §101
yalues of x between a and h* The coefficients a,- will be in gen-
eral unknown, but they do not occur in the calculation, as we shall
see. Let x©, x^, ••, ar^-i be values of x between a and 6, and let
<^(x) be a polynomial of degree n — \ which assumes the same
values as does f{x) for these values of x. Then Lagrange's inter-
polation formula shows that this polynomial may be written in the
form
jii-i
*(*) = X ''-^'-(^) ■*■ ^a.(a^0)^0 W + • • • + ^2,.(^»-l) *n-l W,
where ^^ and 4^^ are at most polynomials of degree n — 1. It is
clear that the polynomial <^,„(x) depends only upon the choice of
3Co> ^i»"*> ^11 -1- ^^ ^'^ other hand, this polynomial <^«(ic) must
assume the same values as does ic"* ioY x — x^^ x — x-^, • • •, x — x^_^.
For, supposing that all the a's except a^ and also Tt^^^ix) vanish,
f{x) reduces to o-^x"* and f\t{x) reduces to a^i^^{x). Hence the
difference x"* — «/>«(«) must be divisible by the product
P^{x) = {x- Xo) (x- Xi) ■■■ (x- iC„_i).
It follows that X"* — <f>„,(x)= P„Q,n_,,(x), where Q,„_„(aj) is a poly-
nomial of degree m — n, if m > 7^ ; and that x"" — (f>j^(x) = it m <n — 1.
The error made in replacing J^ f(x) dx by J^ <^ {x) dx is evidently
given by the formula
(37) Va, I [X- - ^^{x)\ dx + / R,,(x)dx
n-1 ^6
t = *Ja
The terms which depend upon the coefficients a^, a-^, • • •, «-„ _i vanish
identically, and hence the error depends only upon the coefficients
^mt ^u-^M "} ^ti-\ ^^d *he remainder R2„(x). But this remain-
der is very small, in general, with respect to the coefficients
^nt ^'n + n •••> ^2«-i- Hence the chances are good for obtaining a
high degree of approximation if we can dispose of the quantities
*o» ari, •••, x,_, in such a way that the terms which depend upon
"•» *■ + !» •••» ocu-i also vanish identically. For this purpose it is
necessary and sufficient that the n integrals
X*> /»«• •»&
• ThU Is A property of any function which is continuous in the interval (a, 6),
to a theorem due to Wolorutrass (see Chapter IX, § 199).
IV, 1102] APPROXIMATE EVALUATION 201
should yanish, where Q< is a polynomial of degree i. We have
already seen (§ 88) that this condition is satisfied if we take P. of
the form
It is therefore sufficient to take for x^, x^, -", x..| the n roots of
the equation /'. = 0, and these roots all lie between a and b.
We may assume that a = — 1 and b = + 1, since all other caies
may be reduced to this by tlie substitution x = (b -\- a) / 2 -\-t(h — a) /2.
In the special case the values of ar^, Xj, •••, x,_, are the roots of
Legendre's polynomial A\. The values of these roots and the
values of A', for the formula (36), up to n = 5, are to be found to
seven and eight places of decimals in Bertraud's Traite de Caleul
integral (p. 342).
Thus the error in Gauss' method is
f Ru{')dx-'^R^{x,) f\(x)dx,
where the functions ^.(ar) are independent of the given integrand.
In order to obtain a limit of error it is sufficient to find a limit of
R^(x)t that is, to know the degree of approximation with which
the function f(x) can be represented as a polynomial of degree
2n — 1 in the interval (a, h). But it is not necessary to know
this polynomial itself.
Another process for obtaining an approximate numerical value of
a given definite integral is to develop the function /(or) in series and
integrate the series term by term. We shall see later (Chapter VIII)
under what conditions this process is justifiable and the degree of
approximation which it gives.
108. Anuler'a pUnimeter. A great many machines have been invented to
measure mechanically the area bounded by a cloeed plane curve.* One of the
most ingenious of these is Amsler^a plauimeter, whose theory affords an interest-
ing application of line integrals.
Let us consider the areas Ai and Aj bounded by the carves described by two
points A I and ^s of a rigid straight line which moves In a plane in any manner
and finally returns to its original position. Let (Xi, yi) and (x^, y,) be the co^r
dinates of the points ^i and .^ti respectively, with respect to a set of rectaugn-
Ur axes. Let / be the distance AiAtn and the angle which Ai At makes with
* A dMcription of these instrumenu is to be foood in a wwk by Abdank-
Abakanowir/ : Ia^ inUffrapKes, la courts inUgmU H ssf oppHoiUion» (Gaathie^
VilUre, 188G).
202 DEFINITE INTEGRALS [IV, §102
the positive x axis. In order to define the motion of the line analytically, Xi, j/i,
and 6 must be supposed to be periodic functions of a certain variable parameter t
which resume the same values when t is increased by T. We have Xa = Xi + i cos^,
ys = Vi + ' sin e, and hence
Xtdyt - y«dxi = Xidyi - j/idxi + Pdd
+ ^(cos^dyi — 8in<?dxi -\-XiCosddd -\-yiBmede).
The areas Ai and Aj of the curves described by the points Ai and ^j, under the
general conventions made above (§ 96), have the following values :
Ai = - fxidyi - yidxu ^2 = J^^V^ ~ V^^-
Hence, integrating each side of the equation just found, we obtain the equation
At = Ai + - Cde + - jcosddyi — am0dxi+ j{xi coBd + yisin tf) dtf ,
where the limits of each of the integrals correspond to the values to and to+ T
of the variable t. It is evident that fdd = 2K7e, where K is an integer which
depends upon the way in which the straight line moves. On the other hand,
integration by parts leads to the formulae
Jxi cos ^d^ = xi sin d — fsin ddxi^
Jyi8medd = — yicoad+ jcosddyi.
But xi sin and yi cos 6 have the same values tor t = to and t = to-\- T. Hence
the preceding equation may be written in the form
As = Ai + Ejtl^ + I icoaddyi — sintf dxi.
Now let a be the length of the arc described by Ai counted positive in a certain
sense from any fixed point as origin, and let a be the angle which the positive
direction of the tangent makes with the positive x axis. Then we shall have
cos^dyi -sin^dxi = (sinacostf - sintfcosa)d« = sinFds,
where V is the angle which the positive direction of the tangent makes with the
potitive direction AiAt of the straight line taken as in Trigonometry. The
pnoeding equation, therefore, takes the form
(88)
Aa = Ai + KnP + iJsmVda.
Similarly, the area of the curve described by any third point At of the straight
line is given by the formula
(«<>) A| = Ai + Kicn + rJsinFds,
li. Eliminating the unknc
, we find the formula
rA, - < A, = (r - t) Ai + K7cU\l - i'),
where r is the disUnce AiAt. Eliminating the unknown quantity /sinFda
totwaea tbase two equations, we find the formula
IV,Jlt«] APPROXIMATE EVALUATION 208
which may be written in the form
(40) Ai(23) + A,(:)l) + A,(12) + Jrjr(l2)(28)(81) = 0,
where (ik) denotes the diHtance between the polnu Ai and il* (<, Jk s 1, S, S)
ukeii witli itii proper sign. A« an appUoation of this formuia, let na oooalder
a etraight line .^i^s of length (a + 6), whose extreroitiee Ax and At deaeribe the
•ante doaed convex curve C. The point ^Ig, which divkiea the line into aeg-
menti of length a and 6, describes a closed curve C which lies wholly indde C.
In this case we have
A,= A|. (12) = o + 6, (28) = -6, (81) = -a, ir = l;
whence, dividing by a + 6,
Ai — At = xab.
But Ai - At iB the area between the two curves C and C. Hence this area ia
independent of the form of the curve C. This theorem is due to Holditch.
If, instead of eliminating jBinVda between the equations (88) and (89), we
eliminate Ai, we find the formula
(41) At = A, + Kx(V* _ p) + (r _ f)JsinFd».
Amsler's planimeter affords an application of this formula. Let AiA^At be a
rigid rod joined at At with another ro<l OAt- The point O being fixed, the point
^t« to which is attached a sharp pointer, is made to describe the curve whose i
is sought. The point At then
describes an arc of a circle or '2^ A'^*
an entire circumference, accord-
ing to the nature of the motion.
In any case the quantities As, £*,
Z, V are all known, and the area
At can be calculated if the in-
tegral /sin Vds, which is to be
taken over the curve C\ described
by the point Ax, can be evaluated.
This end .^i carries a graduated ^"^
circular cylinder whose axis coin-
cides with the axis of tlie rod i4ti4s, and which can turn about this
I^et us consider a small displacement of the rod which carries Ax At At Into
the position A'\ A%Ai. I>et (^ be the intersection of these straight linea. About
Q as center draw the circular arc A\a and drop the perpendicular ^(P from
A\ upon AxAf We may imagine the motion of the rod to consist of a sliding
along its own direction until Ax comes to a, followed by a rotation about ^ which
brings alo A{. In the first part of this process the cylinder would slide, with-
out turning, along one of ita generators. In the second part the roution of
the cylinder is measured by the arc aAi. The two ratios aA'x/A'xP and
ill P/arc A\A{ approach 1 and sin V, respectively, as the arc ^(^i approaches
aero. Hence a^( = As (sin V + «), where « approaches zero with As. It follows
that the total rotation of tlie cylinder is proportional to the limit of the sum
£A«(Min r 4- (), that is, to the integral /sinKcCa Henoe the meaauremeDt of
this roUtion is sufficient for ttie determination of the given
204 DEFINITE INTEGRALS [iv, Exs,
EXERCISES
1. Show that the sum 1/n -f l/(n + 1) + • • + l/2n approaches log 2 as n
inoresMS indefinitely.
[Show that this sum approaches the definite integral J^ [1/(1 + x)] dx as its
limit.]
2. As in the preceding exercise, find the limits of each of the sums
n« + l n« + 2!' n2 + (n-l)»
1 +^.i^ + ...+ 1
Vn«-1 Vn2-22 V»2 - (n - 1)2
bj connecting them with certain definite integrals. In general, the limit of
the sum
^'t>{hn),
as n becomes infinite, is equal to a certain definite integral whenever 0(i, n) is
a homogeneous function of degree — 1 in i and n.
3. Show that the value of the definite integral //^^ log sin x dx is
-(jr/2)log2.
[This may be proved by starting with the known trigonometric formula
sm-sm sm^ '— = ,
n n n 2'»-i
or else by use of the following almost self-evident equalities :
WW —
r*logsinxdx = I ^ logcosxdx = - I log/ idx.l
Jo Jo 2j, V 2 / ■■
4. By the aid of the preceding example evaluate the definite integral
IT
Jlx jtanxdx.
i, V 2/
6. Show that the value of the definite integral
dx
r'logOjfx)
Jo 1 + ^"
li(jr/8)log2.
[Set X E un and break up the transformed integral into three parts.]
6*. Sfaluate the definite integral
J log(l- 2acosx + a^(2x.
[POISSON.]
IV. Kxi.] EXERCISES 206
[ Dividing the interval from to ir into n equal parts and applying a well^nown
formula of trigonometry, wa are led to aeek the limit of the eTpwton
?Iog[^(a--I)]
n La + 1 -•
aa n beoomee infinite, if a lies between — 1 and -f 1, this limit la »!ro. If
a« > 1, it is jr log a*. Compare f 140.]
7. Show that the value of the definite integral
i:
slnxdg
VI- 2arcos2 + afl
where oc is positive, is 2 if or < 1, and is 2/a if a > 1.
8«. Show that a necessary and sufficient condition that /(z) ihoald be int»-
f^rable in an interval (a, b) is tliat, corresponding to any preassigned number e«
a subdivision of the interval can be found such that the difference 5 — f of the
corresponding sums 8 and a is less than e.
9. Let/(z) and 0(x) be two functions which are continuousan the interval (cu 6)*
and let (a, Zi, x«, • • , 6) be a method of Subdivision of that interval. If (,, ir^
are any two values of x in the interval (x,_f, x,), the sum ^/{^i) <P{vt'^ (Xi — Xf_i)
approaches the definite integral f^/{x) 0(z) dz as its limit.
10. Let/(x) be a function which is continuous and positive in the interval (a, b).
Show that the product of the two definite integrals
!>''''• £m
\a a minimum when the function is a constant.
11. Let the symbol /'> denote the index of a function (§ 77) between So
and X\. Show that the following formula holds:
where e = + 1 if jyxa] «» anu _nj\} <0, « = — 1 if /(zo) <0 and /(zi)>0, and
« = if /(xo) and /(zO have the same eign.
[Apply the laAt formula in the second paragraph of f 77 to each of the ftino-
Uons/(x) and l//(x).]
12*. Let U and V be two polynomials of degree n and n — 1, reepeetlvely,
which are prime to each other. Show that the index of the rational fraction
V/U between the limits - ao and + oo is equal to the difference between the
number of imaginary roots of the equation V + t'T = in which the ooeflkitont
of i is positive and the number in which the coefficient of t is negative.
[Hbrmitk, BulUiin de la 8oei^ fMiM^maUque, Vol. VII, p. 128.]
13*. Derive the second theorem of the mean for iniegrala by integrmtioii 1^
I»arts.
206 DEFINITE INTEGRALS [IV,Exs.
rLet/(x) and ^(z) be two functions each of which is continuous in the inter-
Tal (a, b) and the ftrat of which, /(z), constantly increases (or decreases) and
bM a oonllnaoui derivaUve. Introducing the auxiliary function
*(z)=fy{x)dx
and integrating by parU, we find the equation
J /(x) 0(«) dx = f(b) *(6) - fHx) *(x) dx .
8lDoe/'(x) always has the same sign, it only remains to apply the first theorem
of the mean for integrals to the new integral.]
14. Show directly that the definite integral fxdy-ydx extended over a
eloMd eontoor goes over into an integral of the same form when the axes are
ropUoed by any other set of rectangular axes which have the same aspect.
15. Given the formula
Jr.6 1
cosXxdx = - (sin X6 - sin Xa),
a X
traloate the integrals
r x»p + ^sinXzdx, fx^PcosXxdx.
16. Let us associate the points (x, y) and (x\ y') upon any two given curves
C and C, respectively, at which the tangents are parallel. The point whose
codrdinates are Xi = px + qx\ yi=py -\- qy% where p and q are given constants,
describes a new curve Ci. Show that the following relation holds between the
co t TM p onding arcs of the three curves :
8i=±p8± qsf.
17. Show that corresponding arcs of the two curves
x = tf\t)-f{t) +0'(«), ^, (x' = r(0-/(«) -<P'{th
V =nt) - Wit) + 0(0, w =nt) + w{t) - 0(«)
hare the same length whatever be the functions /(f) and <f>{t).
18. From a point If of a plane let us draw the normals MPi, • • • , MPn to
a given carves Ci, Cs, • • •, Cn which lie in the same plane, and let U be the
distance MPi. The locus of the points M, for which a relation of the form
F(l|, ttt • • •♦ M = holds between the n distances i,, is a curve r. If lengths
pfoportioaal to tF/dU be laid off upon the lines MPi, respectively, according to
* <liftitte eoBTentlon as to ilgn, show that the resultant of these n vectors gives
tiM dirtetion of the normal to r at the point M. Generalize the theorem for
in
19. l«t C be any cloeed curve, and let us select two points p and p' upon the
U> C at a point m, on either side of m, making mp = mp\ Supposing
that the distance mp varies according to any arbitrary law as m describes the
C, dbow that the points p and p' describe curves of equal area. Discuss
where mp la constant.
i\ , Exs] EXERCI8E8 207
20. Given any closed conTex cunre, let ai draw a panOal eai-ft bj UjiBf oA
a constant length I upon the norm&la to the given cture. Show tliAt Um m«a
bet>^en the two curves U equal to ± ir(> + at, wh«r« • ia the laogtli o( tba
21. Let C be any closed curve. Show that the loena of the poinU A, for
which the corresponding pedal has a constant area, is a circle wboas osai« ii
fixed.
[Take the equation of the curre C in the tangntial form
zcost-f y sine =/(<).]
22. Let C be any closed curve, Ci its pedal with respect to a point A, and C|
the locus of the foot of a perpendicular let fall from A upon a normal to C.
Show that the areas of these three curves satisfy the relation A = Ai - A^.
[By a property of the pedal (§ 36), if /> and m are the polar coordinates of a point
on Ci, the coordinates of the corresponding point of Cs are / and « + r/2, and
those of the corresponding point of C are r = Vf^ + p'* and ^ = m 4- are tan Z*/^.]
23. If a curve C rolls without slipping on a straight line, every point A whkh
is rigidly connected to the curve C describes a curve which is called a rMictts.
Show that the area between an arc of the roulette and its base Is twice the area
of the corresponding portion of the pedal of the point A with respect to C. Also
show that the length of an arc of the roulette is equal to the loDgth of the corre-
sponding arc of the pedal. ro— . i
^BTKIJIBB.J
[In order to prove these theorems analytically, let X and T be the eodidl-
nates of the point A with respect to a moving system of axes formed of the
tangent and normal at a point M on C. Let « be the length of the are OM
counted from a fixed point O on C, and let w be the angle between the tanfSDta
at and M. First establish the formulse
da-\-dX= Ydw, dF + XA# = 0,
and then deduce the theorems from them.]
24*. The error made in Gauss* method of quadrature maj be eipvMnd la
the form
X
2 r I.2.8. -m -1*
+ lLl.J.-(«ii-l)J'
1 . 2 . . . 2n 2n
where { lies between - 1 and +L [Mahsiob, Convtes rtmiuM, 1886.]
CHAPTER V
INDEFnaXE INTEGRALS
We shall review in this chapter the general classes of elemen-
tary fonctioDB whose integrals can be expressed in terms of ele-
owntary functions. Under the term elementary functions we shall
include the rational and irrational algebraic functions^ the exponen-
tial function and the logarithm, the trigonometric functions and
their inrerses, and all those functions which can be formed by a
finite number of combinations of those already named. When the
indefinite integral of a function f(x) cannot be expressed in terms
of these functions, it constitutes a new transcendental function.
The study of these transcendental functions and their classification
is one of the most important problems of the Integral Calculus.
I. INTEGRATION OF RATIONAL FUNCTIONS
103. General method. Every rational function f(x) is the sum of
an integral function E(x) and a rational fraction P(x)/Q(x), where
P{x) is prime to and of less degree than Q{x). If the real and
imaginary roots of the equation Q{x) be known, the rational frac-
tioo may be decomposed into a sum of simple fractions of one or the
oiher of the two types
A Mx-{- N
The fnetions of the first type correspond to the real roots, those
of the eeoond type to pairs of imaginary roots. The integral of
the integral function E(x) can be written down at once. The inte-
grtls of the fractions of the first type are given by the formulae
Adx A
(^^^ — (m^lXx-ar-^' ifm>l;
= ^log(aj-a), if ?;i = l.
for lU take of simplicity we have omitted the arbitrary constant C,
which htlOBfi on the rigUtrhand side. It merely remains to examine
908
/ Adx
X — a
V.$103] RATIONAL FUNCTIONS S09
the simple fractions which arise from pairs of imigioary rtKHa
In order to simplify the corresponding integrmU, let us maW^ ^
substitution
a5 = a + /8f, dx^fidt.
The integral in question then beoomes
J [(^ - «)' + ^"]- " iS"- V (1 + <«)V. ^
and there remain two kinds of integrals :
r tdt r dt
J (1 + 0-' J (1 + ^-*
Since tdt is half the differential of 1 4- ^', the first of these inte-
grals is given, if n > 1, by the formula
/
tdt 1 ff*-«
(l + <0"~ 2(»-l)(l + «')— " 2(«-l)[(z-a)« + ^]'
or, if n = 1, by the formula
The only integrals which remain are those of the type
/;
dt
If n = 1^ the value of this integral is
/
dt ^ ^ ♦ •^-<'
= arc tan ^ = arc tan
l+^» P
If n is greater than unity, the calculation of the integral may be
reduced to the calculation of an integral of the same form, in whioh
the exponent of (1 + t^) is decreased by unity. Denoting the inie
gral in question by /,, we may write
r - C dt __ r i-h^-<» __ r dt _ r ^dt
^--j (i+ty-j (14-0- '"-J (i + o— J(i + <v*
From the last of these integrals, taking
tdt 1
(l + O" 2(fi-l)(l + r«)-»
210 INDEFINITE INTEGRALS [V,§103
and integrating by parts, we find the formula
C i^dt t , _J__ C_Ji__,
JiTT^'' 2(n-l)(l4-^*r-* 2(71-1) J (1+^r-^
Substituting this value in the equation for /^, that equation becomes
,=2^^...+
Repeated applications of this formula finally lead to the integral
/i = arc tan t. Retracing our steps, we find the formula
, (2n-3)(2n-6) .3.1 ^ , , _.,.
^- = (2n-2)(2n-4)...4.2 "^^^""^ + ^(^)-
where R(t) is a rational function of t which is easily calculated.
We will merely observe that the denominator is (1 + t^y~^, and that
the numerator is of degree less than 2n — 2 (see § 97, p. 192).
It follows that the integral of a rational function consists of
terms which are themselves rational, and transcendental terms of
one of the following forms :
log (x — a), log [(x — ay -f- )8*], arc tan
X
Lei OB consider, for example, the integral J[l/(x^ — 1)] dx. The
denominator has two real roots + 1 and — 1, and two imaginary
roots + » and — t. We may therefore write
1 A , B Cx-{- D
x*-l aj-lx + l' l-\-x^
In order to determine A^ multiply both sides by a; — 1 and then set
X = 1. This gives A = 1/4, and similarly B = — 1/4. The iden-
tity assumed may therefore be written in the form
-1— 1/_J L\
x*-l 4Va;-l x + lj
Cx -\-D
l-{-x^
or, simplifying the Jeft-hand side,
- 1 _ Cx + D
2(1 -h«^"" 1+x^'
11 follows that C = and D = - 1/2, and we have, finally,
1 1 1 1__
^.^. **--l 4('-l) 4(x+l) 2(x' + iy
whieh gives
J*_j^ = j,„g(i_|^_i
^arc tana;.
V, §K>4] RATIONAL FLXCTIONS 211
Note. The preceding method, though absoluiely gMieral, it not
always the simplest. The work may often be shorteoed by utinf
a suitable device. Let us consideri for example, the integral
/i
dx
(x«-l)-
If n > 1, we may either break up the integrand into partial frac-
tions by means of the roots + 1 and — 1, or we may uae a rednctioo
formula similar to that for /.. But the most elegant method ia to
make the substitution a: = (1 -f «)/(! — «), which gives
J {x^ - 1)- 4- J «• ""•
Developing (1 — «)*•-* by the binomial theorem, it only remaina
to integrate terms of the form Azf^y where /& may be poeitire or
negative.
104. Hermite's method. We have heretofore supposed tuai uie
fraction to be integrated was broken up into partial fractions, which
presumes a knowledge of the roots of the denominator. The»fol-
lowing method, due to Hermite, enables us to find the algebraio
part of the integral without knowing these roots, and it inrolTea
only elementary operations, that is to say, additions, multiplications,
and divisions of polynomials.
Let f(x)/F{x) be the rational fraction which is to be integrated.
We may assume that /(x) and F{x) are prime to each other, and
we may suppose, according to the theory of equal roots, that Um
polynomial F{r) is written in the form
F(a-) = -Y,X}A';..-.Y;,
where A'l, A,, • • •, A'^ are polynomials none of which have moltiplf
roots and no two of which have any common factor. We may now
break up the given fraction into partial fractions whoae
tors are A'j, A J, • • •, A J :
where .4, is a polynomial prime to A<. For, by the theory of high-
est common divisor. '^^ v mwI i' are any two Txtlvnomiala which are
J12 INDEFINITE INTEGRALS [V,§104
prime to each other, and Z any third polynomial, two other poly-
nomials A and B may always be found such that
BX-^-AY^^Z.
Let us set .Y = Xi, K = A'J • • • JYJ, and Z =/(«). Then this identity
becomes
BX^^-AX\''Xl=^f{x),
or, diriding by F(a;),
It also follows from the preceding identity that if f{x) is prime to
JP(x), A is prime to A'l and B is prime to Z| • • • AJ. Repeating the
process upon the fraction
B
X\"'Xl
and so on, we finally reach the form given above.
It is therefore sufficient to show how to obtain the rational part
of an integral of the form
Adx
f
where ^(x) is a polynomial which is prime to its derivative. Then,
by the theorem mentioned above, we can find two polynomials B
and C such that
B<f>(x)+C<t>Xx) = A,
and hence the preceding integral may be written in the form
If n is greater than unity, taking
And integrating by parts, we get
wheooa, substituting in the preceding equation, we find the formula
/Ajg_ c r A,dx
V,$104] RATIONAL FUNCTIONS 21S
where ^^ is a new polynomial. If n > 2, we idaj apply Uia nm*
process to the new integral, and ao on : the proceea may alwftyi be
continued until the exponent of ^ in the denominator ia equal to
one, and we shall then have an expression of the form
/!?-«<-)-/'?■
where R{x) is a rational function of z, and ^ is a polynomial wboee
degree we may always suppose to be less than that of ^, bat which
is not necessarily prime to <^. To integratt* the latter form we must
know the roots of <^, but the evaluation of tliis integral will intitK
duce no new rational terms, for the decomposition of the fraction
\l//<t> leads only to terms of the two types
A Mx-^-N
x-a (x-a)«-f/J«'
each of which has an integral which is a transcendental function.
This method enables us, in particular, to determine whether tho
integral of a given rational function is itself a rational funetioD.
The necessary and sufficient condition that this should be true is
that each of the polynomials like ^ should vanish when the pi
has been carried out as far as possible.
It will be noticed that the method used in obtaining the
for In is essentially only a special case of the preoeding
consider the more general integral
f,
^ AitO, B^^AC^O,
(^x« + 2Bz + C7)»
From the identity
A{Ax* + 2Bx + C) - (ilx + B)« = AC' B*
it is evident that we may write
r dx ^ A r dz
J (Ax* + 2J?x + C7)« AC'B*J {Aa* + IBs + Cy—«
Integrating the last integral by parts, we find
/
Ax-^B . Am-k-B
^'^'''^^(Ax* + 2Bz + 0-^' nn-lHA^-^tBx-^Cr
-I
A r <»
214 INDEFINITE INTEGRALS [V, § 104
wlHlioe the preceding relation becomes
/ dz Az + B
{Az^ + IBx + O" " 2(n - 1)(^C - B^){Ax^ + 2Bx + C)«-i
2fi - 3 A f dx
^2n-2 AC-B^J {Ax'^ + 2Bx + C^""!
Continuing the same process, we are led eventually to the integral
/
dx
Ax^ + 2Bx-\- C
which is a logarithm if B^ - ^C>0, and an arctangent ilB^-AC<0.
As another example, consider the integral
f.
6x« + 3x - 1 ,
ax.
(X8 + 3X + 1)8
From the identity
6x« + 3x - 1 = 6x(x2 + 1) - (x8 + 3x + 1)
it is erident that we may write
r_6x^+3x-l^^ r 6x(x^ + l) ^^_ r dx
J (z« + 8x + 1)« J (x8 + 3x + 1)« J (x3 + 3x + l)^
Integrating the first integral on the right by parts, we find
r 6(x« + i)dx ^ -X r dx
J (z« + 3x + 1)» (x« + 3x + 1)2 J (x3 4- 3x + 1)^
whence the value of the given integral is seen to be
5x« + 8x - 1
/
dx = — — — •
(x» + 3x + 1)8 (x8 + 3x + 1)2
Note. In applying Hermite's method it becomes necessary to solve the fol-
lowing problem : given three polynomials A, B, C, of degrees m, n, p, respectively^
two qf vthicK, A and B^ are prime to each other, find two other polynomials u and v
miek tkat the reUUion Au + Bv = C is identically satisfied.
In order to determine two polynomials u and v of the least possible degree
which solve the problem, let us first suppose that p is at most equal to m + n — 1.
Than w may take for u and v two polynomials of degrees n — 1 and w — 1,
mptclivsly. The m-\- n unknown coeflBcients are then given by the system of
m -f A linear non-homogeneous equations found by equating the coefficients.
For the determinant of these equations cannot vanish, since, if it did, we could
find two polynomials u and v of degrees n — 1 and m — 1 or less which satisfy
tkt Idaotlty <Ati + Be = 0, and this can be true only when A and B have a
II the da^rM of C Is equal to or greater than m + n, we may divide Chy AB
tod obtain a remainder C whose degree is less than m-\-n. Ttun C = ABQ-\-C%
and« making the aabetitutlon u - BQ = ui, the relation Au-\- Bv = C reduces to
Aut -f Bf B C. This is a problum under th» first case.
V,§103] RATIONA! M VCTI0N8 tl5
105. Integ:ral8 of the type /r(x, VA^ + SbT+C) dx. After th*
integrals of rational functions it is natural to ooosider the inte-
grals of irrational functions. We shall commenoe with the ease in
which the integrand is a rational function of x and the iqaare tool
of a polynomial of the second degree. In this case a simple tubititu-
tion eliminates the radical and reduces the integral to the preceding
case. This substitution is self-evident in case the expression under
the radical is of the first degree, say ax -{- b. If we set ox -f- ^ v <*,
the integral becomes
j r{x, y/ax -f h)dx = f/^r
and the integrand of the transformed integral is a rational ftmetioo.
If the expression under the radical is of the second degree and
has two real roots a and b, we may write
and the substitution
V
X — a ^ Aa — bi*
A 7 = tt or r = r-»
x-b ' .1 - ^
actually removes the radical.
If the expression under the radical sign has imaginary roots, the
above process would introduce imaginaries. In o rder to get to the
bottom of the matter, let y denote the radical V^x* -f 2Bx + C
Then x and y are the coordinates of a point of the curve whose
equation is
(1) y« = Ax* + 2Bx -f- C,
and it is evident that the whole problem amounts to expressing liie
coordinates of a point upon a conic by means of rational functions
of a parameter. It can be seen geometrically that this is possible.
For, if a secant
y-/5 = <(x-a)
be drawn through any point (a, fi) on the conic, the coordinates of
the second point of intersection of the secant with the oooic are
given by equations of the first degree, and are therefore rational
functions of t.
If the trinomial Ax* + 2Bx + C has imaginary roots, the coefl-
cient .4 must be positive, for if it is not, the trinomial will be
negative for all real values of x. In this case the conic (1) is an
216 INDEFINITE INTEGRALS [V,^i06
hyperbola. A straight line parallel to one of the asymptotes of
this hyperbola,
y=zxy/A-\-t,
enU the hyperbola in a point whose coordinates are
C -t* r- C -f^
If A < 0, the conic is an ellipse, and the trinomial ^x^ + 2Bx + C
must have two real roots a and b, or else the trinomial is negative
for all real values of x. The change of variable given above is pre-
cisely that which we should obtain by cutting this conic by the
moTing secant
y = t(x — a).
As an example let us take the integral
dx
/;
(x^ + k) Va;2 + k
The auxiliary conic y* = a;* + A; is an hyperbola, and the straight line
X + y = ^ which is parallel to one of the asymptotes, cuts the hyper-
bola in a point whose coordinates are
Making the substitution indicated by these equations, we find
or, retoming to the variable x,
/;
dx X — -y/x^ + k X 1
(x« + A:)* A;Vx'» + A; k^/x^-^k k
the right-hand side is determined save for a constant term
In general, if ^4 C — B* is not zero, we have the formula
dx 1 Ax + B
{Ax* + 2Bx + C)* " AC -B* y/Ax^ + 25a; + C
In tome enees it is easier to evaluate the integral directly without
fMnoring the radical. Consider, for example, the integral
dx
/;
/
y/Ax* f 2Bx -f C
V,ilOfi] RATIONAL FI'VCTIONS JIJ
If the coetiicient A is positive, the integral may be writtflii
r ^dx r VAdx
J y/A^x* + 2ABx + AC J ^(Az-^B)*-^ AC ■- B**
or setting Ax -^ B = t,
-7= / . ^ : = "7=log(<-i-V<«^^C-B*).
y/AJ ^t*^AC-B* WA '
Returning to the variable x, we havft the formula
If the coefficient of x^ is negative, the integral may be written in
the form
C dx r y/ldx ^
J V- Ax^ + 25a; + C J y/ AC + B^ -(Ax - B)**
The quantity AC + B'^ is necessarily positive. Henoe, twaUng ^ii^
substitution
Ax- B = t VTcwTb*,
the given integral becomes
\ r dt 1
— = I ■ • = —7= arc sin ^.
Hence the formula in this case is
r dx 1 . Ax-B
I ^ = — ^ arc sin . =•
J V- ^x» + 2fia; + C Va >MC + B«
It is easy to show that the argument of the arcsine Tariee from — 1
to + 1 as X varies between the two roots of the trinomial.
In the intermediate case when A ssO and B ^ 0, the integral i«
algebraic :
/
dx 1
V2Bx -f C ^
Integrals of the type
dx
= - V2Bx + C.
/
(x - a)V.4x« + 2J5X + C
218 INDEFINITE INTEGRALS [V, §100
reduce to the preceding type by means of the substitution x = a-\-l/y
We find, in fact, the formula
r dx ^ _ r dy
J {x-a) V^a5« + 2Bx -f- C J \/ A^'/ -\-2B-^y -^ C^
where
i4i = ^a« -f iBa + C, B^z=Aa-\-By Ci = A.
It should be noticed that this integral is algebraic if and only if
the quantity a is a root of the trinomial under the radical.
Let us now consider the integrals of the type /Va;^ -f A dx. Inte-
grating by parts, we find
f^:^'+Adx = xV^^^TA- f-^£g
J J vvT
dx
. A
On the other hand we have
f^^=fv^^T^dx-f-^
7 V^^Ta J J Var^-f
= / Vx^ H- Adx - A log(a: + Vx^-f-vl).
From these two relations it is easy to obtain the formulae
(2) j^x* + Adx= ^v^M^+|log(xH-V^M^),
^^^ /^Sf^ = |ViM:^-|iog(. + V^rr7^.
The following formulsB may be derived in like manner :
(4) Jv^5Tr^d»= fV^F3p+^
. X
arc sin ->
a
. X
arc sin —
a
Af !jI!* ^'^ ** ^ kyperboU. The preceding integrals occur in the evaluation
Of tlM MM of a MCtor of an ellipse or an hyperbola. Lot us consider, for
Um hyperbola
V, § 108]
UATIOXAI, FI'VCTIONS
219
and let uh try to flii<l ii .- u . ;i oi a gegmeut ^IfP boiUMl«d bjr Um m« AM, tte
X axis, aud the urdiuuic MI'. Tbk Area !■ eqiul to th* <**< yfi iit Inuaml
X
'6
V3?^^<fe,
that is, by the formula (2),
i^[,V^--..o.(i±v^]
But MP = V = (Va) Vz> - a*, and the term {b/2a)x V«*-^ to pradMly Um
area of the triangle OAfP. Hence the area 5 of the aeetor 0AM, boondtd bj
the arc AM and the radii vectores OA
and OM, is
This formula enables us to exproH
the coordinates x and y of a point M
of the hyperbola in terms of the area S.
In fact, from t^e above and from the
equation of the hyperbola, it is easy to
show that
a b
Pio.»
e"i*,
or
The functions which occur on the right-hand side are called the
cosine and 'sine :
coshx
e* + €
ih« =
ff'-e-J
2 S
The above equations may therefore be written in the form
23
X = a cosh — »
ab
y = 6 eiDh
Theee hyperbolic functions possees properties analofona to Ihoee of the trffo>
nometric functions.* It is easy to deduce, for Inetaiico, the foUowiag
cosh'z - sinh*x = 1,
cosh (z H- y) = cosh x coah y + sinh z dnh y,
sinh (x + y) = sinh z cosh y •)- sinh y ooehz.
• A table of the I<>)n»rithm8 of theee funetioae for poelttTe Taloee of the
is to be found iu Houel's Recueii du/ormuiM nmmtriqw$».
sso
INDEFINITE INTEGRALS
[V,§107
bs ibown in like manner that the coordinates of a point on an ellipse
■ij be exprwMed In termi of the area of the corresponding sector, as follows :
28
aco8— •»
ab
. . 25
ftsin— -.
db
la the OM* of a circle of unit radius, and in the case of an equilateral hyperbola
whoM tf*"***^ is one, these formuls become, respectively,
2 = C06 25, y = sin25;
z = coah25, y = sinh2iS.
It li erident that the hyperbolic functions bear the same relations to the equi-
lateral hyperbola as do the trigonometric functions to the circle.
107. RectUcatioB of the parabola. Let us try to find the length of the arc of
A parabola 2py = x* between the vertex O and any point M. The general
fonnula giTW
arc
»'-xv-(s'-r^-
or, applying the formula (2),
arc OM
X Vz« + pg
2^
|,o.(-^fI^
aA
Tbe algebraic term in this result is precisely the length MT of the tangent,
for we know that 0T=: z/2, and hence
V** ^^*1 ,,^ .. . X* X* . X2 X2(x2 + p2)
wn
*^ 4 4pa 4
4p2
If we draw the straight line connecting T to the focus F, the angle MTF will
be a right angle. Hence we
have
FT
=v?
2 X2 ] /
+ - = Wp2-|.x2,
r r'
Fio.23
ll UDgMt at JT' to the X axis. OM' = arc OM.
poiiUoo r such that If'r = MT, and the focus
by laying off T'F' = TF on a line parallel to the y a^is,
X and r of the point F' are then
whence we may deduce a curi-
ous property of the parabola.
Suppose that the parabola
rolls without slipping on the x
axis, and let us try to find the
locus of the focus, which is sup-
posed rigidly connected to the
parabola. When the parabola
The point T has come into a
F is at a point F' which is
The coijrdi-
X«aroOJf-jrr = |log(?±2^^±Z),
V,5108] RATIONAL FUNCTIONS
and the equation of the locus ia giTen by elimlnaUog x bt t i ma Umm two
tioiiH. Fmiii iht* firHt w»< find
to which we may add the equation
_H
X - Vx* + p« = - pe »•
aince the product of the two left-hand aides la eqiuU to - ^. SabCncUiif
two equations, we iind
and the desired equation of the locus ia
IX t.T'
-f(---'-)=l«-T
This curve, which is called the catenary^ la quite 9Uj to oooatmet. Ita fi
is somewhat similar to that of the parabola.
108. Unicursal curves. Let us now consider, in geoenly the inte-
grals of algebraic functions. Let
(6) F(x,y) =
be tlie equation of an algebraic curve, and let R(iy y) be a rational
function of x and y. If we suppose y replaced by one of the roota
of the equation (6) in H(x, y), the result is a function of the single
variable x, and the integral
/
R(Xy y)dx
is called an Abelian integral with respect to the cnrre (6). When
the given curve and the function R(Xf y) are arbitrary these inte-
grals are transcendental functions. But in the particular case where
the curve is unicursal, i.e. when the cxxirdinates of a point on the
curve can be expressed as rational functions of a variable param-
eter t, the Abelian integrals attached to the curre can be reduoed at
once to integrals of rational functions. For, let
be the equations of the curve in terms of the parameter t Taking
t as the new independent variable, the integral becomes
fR(x, 1/) dx =Jr lJ[t), ^(t)']f{t)di,
and the new integrana is evidently ritiiollsL
2SS INDEFINITE INTEGRALS [V,§108
It is shown in treatises on Analytic Geometry * that every uni-
ennai curve of degree n has (n — l)(n - 2)/ 2 double points, and,
ooQTersely, that every curve of degree n which has this number of
doable points is unicursal. I shall merely recall the process for
obtaining the expressions for the coordinates in terms of the param-
eter. Given a curve C. of degree n, which has S = {n — l)(n — 2)/2
double points, let us pass a one-parameter family of curves of degree
•I — 2 through these S double points and through n — 3 ordinary points
on C,. These points actually determine such a family, for
(n-l)Cn-2) , ^^ 3_ (n-2)(n+l) ^^
2 2
whereas (n — 2)(n -f l)/2 points are necessary to determine uniquely
a curve of order n — 2. Let P(x, y) -f tQ{Xy y) = 0\)Q the equation
of this family, where ^ is an arbitrary parameter. Each curve of the
family meets the curve C, in n(n — 2) points, of which a certain num-
ber are independent of t, namely the n — 3 ordinary points chosen
above and the 8 double points, each of which counts as two points of
intersection. But we have
n - 3 -f 28 = n - 3 -f (w - l)(7i - 2) = 7i(7i - 2) - 1 ,
and there remains just one point of intersection which varies with t.
The coordinates of this point are the solutions of certain linear equa-
tions whose coeflBcients are integral polynomials in t, and hence they
are themselves rational functions of t. Instead of the preceding we
might have employed a family of curves of degree n — 1 through the
(n — l)(n — 2)/2 double points and 2n — 3 ordinary points chosen at
pleasure on C..
If n s= 2, (n — l)(n — 2)/2 = 0, — every curve of the second
degree is therefore unicursal, as we have seen above. If w = 3,
(» — l)(n — 2)/2 = 1, — the unicursal curves of the third degree
are those which have one double point. Taking the double point
M origin, the equation of the cubic is of the form
<^«(^»y)4-</»,(a;, 2/) = 0,
where ^, and ^ are homogeneous polynomials of the degree of their
indioee. A secant y = tx through the double point meets the cubic
in a single variable point whose coordinates are
^•a»0 "^ Mht)
•Stt, •.§., NI«WMflowild, Court de Q4omitH« ancUytiqm., Vol. II, pp. 9&-114.
V,fl08] UATIONAL FUNCTIONS Sf|
A unicursal curve of the fourth degree hai three doable points.
In order to find the coc^rdinates of a point on it, we ihookl paee a
family of conies through the three double points and through anoUMr
point chosen at pleasure on the curve. Every conic of this familj
would meet the quartic in just one point which Taries with the
parameter. The e(|uation which gives the sbeeissaD of the points of
intersection, for instance, would reduce to an equation of the first
degree when the factors corresponding to the doable points hsd
been removed, and would give x as a rational fnnetion of the
parameter. We should proceed to find y in a similar manner.
As an example let us consider the lemniscate
(aj« + y«)« = «'(x»-/),
which has a double point at the origin and two others at the imagi-
nary circular points. A circle through the origin tangent to one of
the branches of the lemniscate,
x« + y» = f(x-y),
meets the curve in a single variable point. Combining these two
equations, we find
or, dividing by x — y,
This last equation represents a straight line through the origin which
cuts the circle in a point not the origin, whose coordinates ars
*- t'j^a' ^" <* + a«
These results may be obtained more easily by the following
process, which is at once applicable to any unicursal curve of the
fourth degree one of whose double points is known. The secant
y — Xz cuts the lemniscate in two points whose coordinates are
X =
14- X«
y-X«.
The expression under the radical is of the second degree. Hsne^
by § 105, the substitution (1 - X)/(l + X) - (a/f)« removes the radi-
cal. It is easy to show that this substitution leads to the
just found.
224 INDEFINITE INTEGRALS [V,§109
NoU, When a plane curve has singular points of higher order, it
oaa be shown "tfcat each of them is equivalent to a certain number of
i^^l^i^ double points. In order that a curve be unicursal, it is suffi-
eient that its singular points should be equivalent to {n — l){n — 2)/2
itolated double points. For example, a curve of order n which has
a multiple point of order n — 1 is unicursal, for a secant through
the multiple point meets the curve in only one variable point.
109. Integrals of binomial differentials. Among the other integrals
in which the radicals can be removed may be mentioned the follow-
ing types:
/ r\x, {ax + hY\dx, I R{x, -s/ax -f- h, Vex -\-d)dXy
R(x'j x"', x*", ■•)dx,
I'
where R denotes a rational function and where the exponents
a, a\ a"f ••• are commensurable numbers. For the first type it is
sufficient to set aa; -f i = ^. In the second type the substitution
€uc -\- b = t* leaves merely a square root of an expression of the
•eoond degree, which can then be removed by a second substitution.
Finally, in the third type we may set x = t^, where D is a common
denominator of the fractions a, a', a", • • •.
In connection with the third type we may consider a class of
di£Ferentials of the form
x''(aaf' + bydxj
which are called binomial differentials. Let us suppose that the
three exponents w, n, p are commensurable. If jt> is an integer, the
expression may be made rational by means of the substitution
' = ^» ae we have just seen. In order to discover further cases
of integrability, let us try the substitution ax"" + b = t. This gives
The transformed integral is of the same form as the original, and
the exponent which takes the place of p is (m + l)/n - 1. Hence
the integration can be performed if {m + l)/n is an integer.
V,§109J BATIONAL FUNCTIONS
On the other hand, the integral may be writteo in the form
f-
whence it is clear that another case of integrabilitj is that in whieh
(w + np -f l)/n = (m + l)/n -f p is an integer. To sum up, tlM
integration can be performed whenever one of the three nmmhere
Pj (m -f l)/n, (m + l)//i -^-p is an integer. In no other case eaa the
integral be expressed by means of a finite number of elementarj
functional symbols when m, n, and p are rational.
In these cases it is convenient to reduce the integral to a simpler
form in which only two exponents occur. Setting a«* a bif we find
' l/6\-i-i
x = l- r, cfa = Mf <• dt,
n \a
x'^(ax''-{-bydx = -l-] "It* 'f^tydi.
Neglecting the constant factor and setting g = (m + l)/n — 1, we
are led to the integral
/
t^(i'^tydt.
The cases of integrability are those in which one of the three m
bers p, q, p -\- q w an integer. If /) is an integer and q = r/«, we
should set t = t/*. If y is an integer and p = r/«, we should set
1 4- < = u\ Finally, if jd -f- y is an integer, the integral may be
written in the form
/'"•(¥)'-.
and the substitution l-{- f= Ui'y where p = r/#, remores the radical
As an example consider the integral
/
xy/l-k-T^dx.
Here m = 1, n = 3, ;> = 1/3, and {m + V)/n +p - 1. Heooe thii
is an integrable case. Setting x* = r, the integral b eoo nm
and a second substitution 1 4- f = /u* remoTes the r adio al.
INDEFINITE INTEGRALS [V,§110
II. ELLIPTIC AND HYPERELLIPTIC INTEGRALS
UO. Reduction of integrals. Let P{x) be an integral polynomial
of degree p which is prime to its derivative. The integral
JR[x,Vp{^)]dx,
where R denotes a rational function of x and the radical y = -\/P(x),
eannot be expressed in terms of elementary functions, in general,
when p is greater than 2. Such integrals, which are particular
etMt of general Abelian integrals, can be split up into portions which
rwalt in algebraic and logarithmic functions and a certain number
of other integrals which give rise to new transcendental functions
which cannot be expressed by means of a finite number of elemen-
tary functional symbols. We proceed to consider this reduction.
The rational function /2(x, y) is the quotient of two integral
polynomials in x and y. Replacing any even power of y, such as
y**» by [.^{^yfi and any odd power, such as y^' + S by y [P(«)]S we
may evidently suppose the numerator and denominator of this frac-
tion to be of the first degree in y,
-,, . A-\- By
R(x, y)= — ■ — ^,
where Aj B, Cy D are integral polynomials in x. Multiplying the
numerator and the denominator each by C — Dy, and replacing y^
by P(ar), we may write this in the form
^(*. y) = —j^>
where F, G, and K are polynomials. The integral is now broken
up into two parte, of which the first Jf/K dx is the integral of a
rationil function. For this reason we shall consider only the second
integral fOy/K dx, which may also be written in the form
/;
Mdx
n-Vp(x)
where If and AT are integral polynomials in x. The rational frac-
tion M/N may be decomposed into an integral part E(x) and a
ram of partial fractious
V, §ii()j hhhirilL ASU llll'tKKLUPTIC INTKliRALS 227
where eai?h of the polynomials A\ is prime to its derivative. We
shall therefore have to consider two types of integrali,
^x) J A'-V7xi)
If the degree of P(x) is p^ all the integraU K. may he exprtsied
in terms of the first p - 1 of them, Ko, K,, •••, K^.,, and certmim
algebraic expressions.
For, let us write
P(x) = aoJB' H- ajaj"-* + • • ••
It follows that
:^ (x- vTy^) = mar- - » Vi^) -H i^^^^^l
^ 2wjr— »P(g) 4- afP*(a?)
2 VP(x)
The numerator of this expression is of degree m +/' — 1, and its
highest term is (2m -\- p)anX'*-*-p-K Integrating both sides of the
above equation, we find
2x'-VP{x)=(2m-^p)a,r^^,.^ + .-•,
where the terms not written down contain integrals of the type
y whose indices are less than m -{- p — 1. Setting m = 0, 1, 2, • • •,
successively, we can calculate the integrals K,_i, K,, ••• suooes-
sively in terms of algebraic expressions and the /> — 1 integralf
With respect to the integrals of the second type we shall distin-
guish the two cases where A' is or is not prime to P(x).
1) If X is prime to P(x), the integral Z^ reduces to the sum of
an algebraic term, a number of integrals of the type K*, and a
integral
Bdx
/:
XWP(x)
where B is a polynomial whose degree is less thorn that of X.
Since -Y is prime to its derivative X' and also to P(ap), JT* is prims
to PX'. Hence two polynomials X and ft can be found such that
XX^ -^ fiX'P = A, and the integral in question bfsaks up into
two parts:
J A-\ /\x) J \/P{x) J
INDEFINITE INTEGRALS [V, §110
The first part is a sum of integrals of the type Y. In the second
integral, when » > 1, let us integrate by parts, taking
which gives
r ^Vpx'dx ^ -mVp ^ 1 CML±JtELdx
J X* (n-l)A"-' n-lJ 2X^-'Vp(^
The new integral obtained is of the same form as the first, except
ihftt the exponent of X is diminished by one. Repeating this
prooees as often as possible, i.e. as long as the exponent of X is
greater than unity, we finally obtain a result of the form
/ Adx r Bdx r cdx dVp
X*y/P(x) J aVp J Vp Z"
where B, C, D are all polynomials, and where the degree of B may
always be supposed to be less than that of X.
2) If X and P have a common divisor Z), we shall have X — YD,
P ra SD, where the polynomials D, S, and Y are all prime to each
other. Hence two polynomials X and fi may be found such that
A = X/)* -I- fiY", and the integral may be written in the form
r Adx ^ r Xdx Tju
J x*Vp J y«Vp J z)"
dx
Vp
The first of the new integrals is of the type just considered. The
9 9oemd i$UegrcLly
/fjL dx
ly'Vp
whmrt D is a factor of P, reduces to the sum of an algebraic term
and a number of integrals of the type Y.
For, since />• is prime to the product D'S, we can find two poly-
BomiaU Aj and fi^ such that \,JJ^ -f fi^D'S = fi. Hence we may write
J iryTp J y/p^J d^Vp"^^'
Replaeing P by DS, let us write the second of these integrals in the
V.§110] ELLIPTIC AND HYPERELLIPllt JMLcltM > 229
and then integrate it by parts, taking
which gives
J L^y/P J V? (n - !)/>•-». 2n-l J ir^VP
This is again a reduction formula ; but in this case, since the expo-
nent n — 1/2 is fractional, the reduction may be performed erao
when D occurs only to the first power in the denominator, ^^ we
finally obtain an expression of the form
dx kVp
J r^y/p />• J V? '
where H and K are polynomials.
To sum up our results, we see that the integral
/
Mdx
nVp
can always be reduced to a sum of algebraic terms and a number of
integrals of the two types
J V^' J X^
dx
Vp
where m is less than or equal to /> — 2, where A' is prune to lU
derivativ.e A" and also to P, and where the degree of A'l is lest thao
that of X. This reduction involves only the operations of aAlifiii,
multiplication, ami division of polynomials.
If the roots of the equation A' = are known, each of the rational
fractions Ai/X can be broken up into a sum of partial fractioni of
the two forms
A Bx-^C
x — a (« — a)* + /8**
where i4, B, and C are constants. This leads to the two new tjpee
C)<£ar
r dx r (Bx -f
J (x - a)VP(^' J [(x - ay +
which reduce to a single type, namely the first of theee, if we
to allow a to have imaginary values. Integrals of this tort ere
280 INDEFINITE INTEGRALS [V, §110
ealled imUgraU of the third kind. Integrals of the type F„, are
eiUed vUegmU of the first kind when m is less than p/2 — 1, and
are called integrals of the second kind when m is equal to or greater
than p/2 — 1. Integrals of the first kind have a characteristic
property, — they remain finite when the upper limit increases
indefinitely, and also when the upper limit is a root of P(x)
(SS S^» dO); but the essential distinction between the integrals of
the leoond and third kinds must be accepted provisionally at this
time without proof. The real distinction between them will be
pointed out later.
Note. Up to the present we have made no assumption about the
degree p of the polynomial P{x). If p is an odd number, it may
always be increased by unity. For, suppose that P(x) is a poly-
nomial of degree 2q —1:
P(x) = ^oa^'-' + A.x^''-' -}-... + A^_,.
Then let us set a; = a -f- 1/y, where a is not a root of P(x). This
giTes
where Pj (y) is a polynomial of degree 2q. Hence we have
and any integral of a rational function of x and VP(ar) is trans-
formed into an integral of a rational function of y and VW^).
ConTersely, if the degree of the polynomial P{x) under the radi-
cal U an even number 2y, it may be reduced by unity provided a
root of P(x) is knoum. For, if a is a root of P(x), let us set
« » a + 1/y. This gives
P(x) = P'(«)- + ... + ^^^^J- = ^lIl^,
y (2q)l y'" y*« '
whara />|(y) U of degree 2y - 1, and we shall have
a«ca the integrand oQhe transformed integral will contain no
dher radical tlian V/»i(y).
V, §111] ELLIPTIC AND IIYPKRKI.IJPTir TXTFORaI^ 281
111. Case of integration m algebraic termi. We Mve jun Men UxMl An UUtgitl
of the form
fRl
X. v7^i)]dx
can always be reduced by mewaa of elementary openitloM to the iOB <d to IbI»-
gral of a rational fraction, an algebraic expreeiion of tbe form O VP^)/L, tad
a number of integrals of the first, second, and third kinds. Since we can also
find by elementary operations the rational part of the integral of a ratiooal
fraction, it is evident that the given integral can always be reduced to tbe fom
J/(x, vm]dx = Fix, vF(ij] + r.
where F is a rational function of x and VP(x), and where 7 is a sum of inl^
grals of the three kinds and an integral fXi /Xdx, X being prime to tie deriva-
tive and of higher degree than .Yi. Liouville showed that if the given integrai
is integrable in algebraic terms, it is equal to F[Xf VP(x)]. We dionld tlMre>
fore have, identically,
«[x, VP(i)] = I Kx, V^)]{.
and hence 7=0.
Hence toe can discover by means of multiplicationM and difMonM <ifpolifmamkU»
whether a given integral is integrable in aigebraic terms or noC, ami in earn U it,
the same process gives the value of the integral.
112. Elliptic integrals. If the polynomial P{x) is of the seoond
degree, the integration of a rational function of x and P{x) eao be
reduced, by the general process just studied, to the calculation of the
integrals
r dx r dx
J Vp(x) ' J {x- a)y/P{x)
which we know how to evaluate directly (§ 105).
The next simplest case is that of elliptic integrals, for which l\x)
is of the third or fourth degree. Either of these oases can be
reduced to the other, as we have seen just above. Let P{x) be a
polynomial of the fourth degree whose coefficients are all real and
whose linear factors are all distinct. We proceed to show that
a real substitution can always be found which carries P{x) into a
polynomial each of whose terms is of even degree.
I^t a, b, c, d be the four roots of P{x). Then there eiists an
involutory relation of the form
(7) Ix'«" + Af («' + «") + JV == 0,
INDEFINITE INTEGRALS [^,§112
which in satisfied by or' = a, x" = ft, and by x' = c, a;" = d. For the
OOiAoients L, A/, ^' ueed merely satisfy the two relations
/:a^ + jjf(a + ft)-f iV' = 0,
which ai« eridently satisfied if we take
Ltma-^b^e — df M ^cd — abf N = ab(c -\- d) — cd(a -\- b).
Let a and ^ be the two double points of this involution, i.e. the
rooto of the equation
Lu* + 2Afu + N = 0.
These roots will both be real if
(cd - aby-(a + b - c - d)[ab(c + d)- cd(a + ft)] > 0,
thatia,if
(8) (a -c)(a- d)(b - c)(b -d)>0.
The roots of P{x) can always be arranged in such a way that this
condition is satisfied. If all four roots are real, we need merely
ohc¥)ee a and ft as the two largest. Then each factor in (8) is positive.
If only two of the roots are real, we should choose a and ft as the real
roots, and c and d as the two conjugate imaginary roots. Then the
two factors a — c and a — d are conjugate imaginary, and so are th^
other two, ft — c and b — d. Finally, if all four roots are imaginary,
we may take a and ft as one pair and c and d as the other pair of
conjugate imaginary roots. In this case also the factors in (8) are
conjugate imaginary by pairs. It should also be noticed that these
methodB of selection make the corresponding values of i, M, N real
The equation (7) may now be written in the form
(9) ^^^l^-f ^'-^=0
^^ x^-fi^ X--P \'
If we Mt (« - a)/{x _ ;3) = y, or a; = {Py - a)/iy - 1), we find
Pi(y) i* a new polynomial of the fourth degree with real
whose roots are
g -• g ft — g e— a d — a
a-^fi' b-p* e-p' JZTp
II if eridnt from (9) that these four roots satisfy the equation
V,J112] ELLIPTIC AND HYPEKELLIPTIC INTEGRALS «tt
y' 4- y" = by pairs ; henoe the polynomUl Px(}f) oonUini do ierm
of odd degree.
If the four roots a, by c, d satisfy the equation a-^htse-^d^wt
shall have Z. = 0, and one of the double points of the inrolutioD lie*
at infinity. Setting a = — N/2My the equation (7) takat the form
rr' — a + as" — a = 0,
and we need merely set x = a -f y in order to obtain a polynomial
which contains no term of odd degree.
We may therefore suppose P(x) reduced to the ^nffnii?a1 form
P(x) = ^o«* + iijX* + At.
It follows that any elliptic integral, neglecting an algebraio term
and an integral of a rational function, may be reduced to the ■am
of integrals of the forms
/dx r xdx r
V%x*T^47xH^' J V^o«*+^iaf*+^«' J V^
and integrals of the form
dx
!-.
(x - a) \MoX* + ^iX« -f -4,
The integral
dx
•■=r-
is the elliptic integral of the first kind. If we consider Xy on the
other liand, as a fuuction of t/, this inverse function is called an
elliptic function. The second of the above integrals reduces to an
elementary integral by means of the substitution ac* = ti. The third
integral
r ^dx
J V.4o«*-|-i4iX«-f i4,
is Legendre's integral of the second kind. Finally, we have tba
identity
r d^ =r '^^ +af ^ ^ .
J (X - a) Vp(i) J (x« - a«)>/P(x) •/(«•- a«) VP(x)
The integral
r dx
J (x« + A)Vi4pX* + i4iX« + ^,
is Legendre's integral of the third kind.
{(4 INDEFINITE INTEGRALS [V, § ns
TheM eUiptic integrals were bo named because they were first
met with in tl.e problem of rectifying the ellipse. Let
x = aco8^, 7/ = b8m<l>
be the codidinates of a point of an ellipse. Then we shaU have
ds»^dx^^dy'-(a^ sin^</» + b^ cos^*^) d<l>\
or, setting a* — A* = c*a*,
ds = aVl — e'cos*<^ dtf*-
Hence the integral which gives an arc of the ellipse, after the sub-
stitution cos ^ = ^ takes the form
s= a I — , dt = a I ' , „^ ^
==dt.
It follows that the arc of an ellipse is equal to the sum of an inte-
gral of the first kind and an integral of the second kind.
Again, consider the lemniscate defined by the equations
An easy calculation gives the element of length in the form
d^^d^^-dy"^ -^-—l dt\
t ~\~ CL
Hence the arc of the lemniscate is given by an elliptic integral of
the first kind.*
lit. PMado-«lUptic integrals. It sometimes happens that an integral of the
form /F[x, VP(x)]dx, where P(x) is a polynomial of the third or fourth
degree, can be expressed in terms of algebraic functions and a sum of a finite
of logarithms of algebraic functions. Such integrals are called pseudo-
Thie hftppens in the following general case. Let
(10) Lx'x" + M{x' + X") + iV =
b$ on hnohiiory relation which eatablishea a correspondence between two pairs of
tktfomr rooU of the qttartic equation P(x) = 0. If the function f{z) be such that
tktrdaUtm
i§ idtntkaUy ioti^fied, the inUffral /[/(x)/ VP(x)] dx is pseudo-eUiptic.
* TlUi to e eommoD property of a whole class of curves discovered by Serret
(Onm 4f C'o/cu/ diffirentUl 9t integral, Vol. II, p. 264).
V.JlKiJ ELLIPTIC A.NiJ ail-hlUuLLil'llL IMtCiUAI^
Let a and /9 be thu double puinu of the iDTOlalloil. Am w« tep
seen, the equation (lOt may be written in the fonn
(12) ?1Z^ + 51ZJU0.
Let u« now make the lubetitution (x — a)/{x - fi)sy. This fl?«
" (l-V)' " (!-»)♦'
and consequently
dg _ {a — fi)dy
where Pi (y) is a polynomial of the fourth degree which contain* no odd powen
of y (§ 112). On the other hand, the rational fraction /(z) goes over into a
rational fraction <p{y), which satisfies the identity ^{y) + ^<- y) = 0. For If
two values of x correspond by means of (12), they are transformed Into two
values of y, say y' and y", which satisfy the equation f' -¥ u" = 0. It iaoridaBt
that <p{y) is of the form y^{y^), where ^ is a rational function of y*.
the integral under discussion takes the form
/
y^iy*)dy
y/AQy*-^Air/* + At
and we need merely set y* = zin order to reduce it to an elementary integral.
Thus the proposition is proved, and it merely remains actually to carry out
the reduction.
The theorem remains true when the polynomial P{x) is of the third degree,
provided that we think of one of its roots as infinite. The d ei B O M U atkm is
exactly similar to the preceding.
If, for example, the equation P{x) = is a reciprocal equation, one of the
involutory relations which interchanges the roots hy pairs is x'z^ = 1. Henee,
if /(x) bo a rational funct ion which satisfies the relation /{x) + /(1/x) = 0,
the integral /[/(x)/VP(x)] dx is peeudo^lliptic, and the two subsUiulioos
(x — l)/(x -f 1) = y, y2 = z, performed In order, transform it into an elementary
integral.
Again, suppose that P(x) is a polynomial of the third
P(«) = x(x
■,(.-i)
Let us set a = 00, 6 = 0, c = 1, d = l/lfi. There exist three Involutory rela-
tions which interchange these roots by pairs :
fc«x" *«(i-o i-*«jr
Hence, if /(x) be a rational function which satisilee one of the U
986 INDEFINITE INTEGRALS [V,§114
f{x)dx
f
Vx{l-x){l-k^x)
Is pModfKeUipUc. From this others may be derived. For instance, if we set
s = I*, the pnoeding integral becomes
/
2f{z*)dz
y/{l-z^){l-k^z^)
whenea it follows that this new integral is also pseudo-elliptic if f{z^ satisfies
OM ol U»e identities
The flrat of these cases was noticed by Euler.*
III. INTEGRATION OF TRANSCENDENTAL FUNCTIONS
114. Integration of rational functions of sin x and cos x. It is well
known that sinx and cosx may be expressed rationally in terms
of tan x/2 = t. Hence this change of variable reduces an integral
of the form
/
i? (sin ic, cosa;)<^a;
to the integral of a rational function of t. For we have
« = Zarctan^ (fo = j-— , sina; = j— ^, cosx = ^— -^>
and the given integral becomes
where ♦(^) is a rational function. For example,
r dx cdt ,
-. — = log tan jiA
/
Binx
• Hoe II«rmit«'s lithographed Courn, 4th ed., pp. 25-28.
V,§114] TRANSCENDENTAL FUNCTIONS S|7
The integral / [l/cos x] dx reduces to the preceding bj mMni of the
substitution x = 7r/2 — y, which gives
/^=-'o«-(M)='<.-(M)-
The preceding method has the advantage of ganendity, hot it is
often possible to find a simpler substitution which is eqnallj suc-
cessful. Thus, if the function /(sin a;, cos x) has the period w, it is
a rational function of tan a;, F(tan x). The substitution tan x s i
therefore reduces the integral to the form
/,(.«. .,^./.!m'.
As an example let us consider the integral
dx
I:
A cos* a; + B sin a; cos X + C sin* x + />*
where A, B, C, D are any constants. The integrand evidently hsM the
period ir ; and, setting tan x = ^, we find
t . . C
Hence the given integral becomes
iT?' smxcosx = j-j-y,, sin«x=:j-^
/:
dt
The form of the result will depend upon the nature of the
of the denominator. Taking certain three of the ooeffieients sero^
we find the formulae
/dx C dx
— ^ = tanx, \- =logtanx,
cos'x J sin X COS X
/;
dx
= — cot X.
sin"x
When the integrand is of the form R(%\tl x) cos x, or of the form
/^(cosx) sinx, the proper change of variable is apparent In the
first case we should set sin x = ^ ; in the second case, ooec ■* I.
It is sometimes advantageous to make a first substtliitioo in older
to simplify the integral before proceeding with the geDeral method.
For example, let us consider the integral
/
dx
acosx4-6 8inx + e
2S8 INDEFINITE INTEGRALS [V, § 114
whew a, 6, e are any three constants. If p is a positive number
and ^ an angle determined by the equations
we shall have
and the given integral may be written in the form
/ dx _ r dy
pco8(x-<^)+c J p cosy -he'
where x — ^ = y. Let us now apply the general method, setting
tan y/2 = t. Then the integral becomes
2dt
f.
P + c + (c- p)t^
and the rest of the calculation presents no difficulty. Two different
forms will be found for the result, according as p^ — c^ = a^ -\-b^— c^
18 positive or negative.
The integral
" m cos X -{- n sin x +p j
; — : ax
a cos X -{- sm x -\- c
f
may be reduced to the preceding. Eor, let u = a cos x -\- h sin x -\- c,
and let us determine three constants X, p., and v such that the equation
du
m cos X + n sin x-\-p = \u-{-p.- — hv
it identically satisfied. The equations which determine these num-
bers are
m = Xa + pjbf n = \b — pa, p = \c -{- v,
the first two of which determine X and p.. The three constants hav-
ing been selected in this way, the given integral may be written in
the form
/
du
dx = Xx -\- pilog u -h V I
«* J a cos X H- i sin X -H c
Let OB try to evaluate the definite integral
dx
£
l-)-fOOS»
« where |e|<l.
vr,§iifi] TRANSCENDENTAL FUNCTIONS 2S9
Cooiidering it llnl •■ an indefinite integnd, we find ■nocuwlnlj
f ^ ^2C ^ -r > C **
Jl + 6ooe« Ji + e + (l-«)<« Vl - > J 1 + 1^'
by means of the sttoceeeiye subetitutione tanx/SsC, (ss«V(l-|- e)/(i > i^
Hence the indefinite integrml ia eqoal to
vT-^
arc tan
m-t)
As z varies from to ;r, V(l - e)/(l + e) tan x/2 ineraMet from to 4* •, aad
the arctangent varies from to ie/2. Hence the gifMi <*^*^««tt liiiif,nl Is afMl
to 7r/V(l - c'-O-
115. Reduction formulae. There are also certain classes of intograls
for which reduction formulaB exist. For instaooey the fonnnla lor
the derivative of taii*"*x may be written
d_
dx
whence we find
— (tan"->x) = (n -1) tan-«a5(l+ tan«x),
/tan«-»x r , ^
tan*x<w;= I tan"""a;ttar.
The exponent of tan x in the integrand is diminished by two units.
Kepeatecl applications of this formula lead to one or the other of
the two integrals
jdx = x, I tanxr/xss— logcosx.
The analogous formula for integrals of the type /cot" xdlx is
/cot"~*j; r
coV'xdx = I cot"""x<ix.
In general, consider the integral
8in"a;co8"a;<ix,
/■
where m and n are any positive or negative integers. Wheo one of
these integers is odd it is best to use the change of vmriable ghreo
above. If, for instance, n = 2;) -f 1, we should set sinx s (, whieh
reduces the integral to the form /<"(!— <*)'<ft.
Let us, therefore, restrict ourselves to the case where m and n are
both even, that is, to integrals of the type
f ^=s I sin'^xooshadb,
240 INDEFINITE INTEGRALS CV,§116
which may be written in the form
/ = / sin'^'^ascos^'ajsincccfe.
Taking oo«»"x sinxdx as the differential of [- l/(2n 4-l)]cos2»+ia;,
an integration by parts gives
/^,= -riB--.^ + ^/sin--xcos-x(l-8in».)<i.,
which may be written in the form
_ 8in**'"^agco8*'"'"^g 2m— 1 j
(-^J i-.. ""■" 2(m + w) 2(m + w) ^«-i.«*
This fonnula enables us to diminish the exponent m without alter-
ing the second exponent. If m is negative, an analogous formula
may be obtained by solving the equation (A) with respect to I^a-x^n
and replacing m by 1 — m :
^^ -'—.."" l-2m "^ l-2m ^i-«,n'
The following analogous formulae, which are easily derived, enable
U8 to reduce the exponent of cos x \
J __ sin*"'-*-^xcos'"~^a; 2n—l j
^' -*".• " 2(m-fn) 2(m + w) ^m.— i'
/D^ /" - - «^'^*"'"^^^<^Q8^"'*^ , 2(m + l-n) ^
^^' ^-.-." i_2» "^ i-2n ^-.-n+r
Repeated applications of these formulae reduce each of the num-
bers m and n to zero. The only case in which we should be unable to
proceed is that in which we obtain an integral /„ ,j, where m + n = 0.
But such an integral is of one of the types for which reduction for-
mula were derived at the beginning of this article.
lit. WalUt' formnUB. There exiat reduction formulas whether the exponents
m and n are •reo or odd.
Am an example let lu try to eyaluate the definite integral
w
/»= r*8ln«"zda;,
whM« m U a poaltlTe integer. An integration by parts gives
J aln— »«iin«d«»-[oos»sin"-»x]» + (m-l) r'8in"-5jCos«xda;,
V,$117J TRANSCENDENTAL FUNCTIONS S41
whence, noting that coex sin^-^z Taniahat at boib »■«<«#, wa fln4 tha fooNlA
w
/« = (m - 1) pain— »x(l - ^n*x)dx = (m - !)(/— t - /^,
which leads to the recurrent formula
(13) /, = !!^^7,_,.
m
Repeated applications of this formula reduce the given integral to /© = r/a
if m is even, or to Ii = \ if m ia odd. In the former caM, taking m - 2p aiid
replacing m succeasively by 2, 4, 6, • • • , 2p, we find
It --Jo, A = -/i, , ^«p= -J— ^S,-lf
or, multiplying these equations together,
1.8.6...(2p-l) jr
'^ 2 . 4 . 6 . . • 2p 2
Similarly, we find the formula
2 . 4 . 6 . . . 2p
2p + l
1 . 3 . 6 . • (2p + 1)
A curious result due to Waliis may be deduced from these formulc It U
evident that the value of /^ diminishes as m incroaaoa, for 8iii"'*'>x la laaa thaa
sin"**. Hence
and if we replace Isp + if hp^ hp-\ by their values from the formula abort, wa
find the new inequalities
2 ' 2p + 1
where we have set, for brevity,
2 2 4 4 2p-2 «p
'^ \'z'^' b"2p-\'%p-\'
It is evident that the ratio ie/2Hp approachea the limit one aa p
nitely. It follows that n/2 is the limit of the product H^ aa tha nombar of
factors increases indefinitely. The law of formation of the snooaarita teotoca ia
apparent.
117. The integral /cos (ax + b) COS (a'x + b')...djL Let as
a product of any number of factors of the form oos(aa; + *X ^*»*«*
a and b are constants, and where the same factor may ocour
times. The formula
C08(M 4- tt) . COS(li - v)
cos u cos t; = ^^ ^ + ;,
242 INDEFINITE INTEGRALS [V,§117
enables us to replace the product of cwo factors of this sort by the
sum of two cosines of linear functions of x ; hence also the product
of n factors by the sum of two products of 71 — 1 factors each.
Repeated applications of this formula finally reduce the given inte-
gral to a sum of the form 1,H cos(Ax + B), each term of which is
immediately integrable. If A is not zero, we have
J'cOB(^:. + J)<fe= ^'"(-^^ + ^> + C,
while, in the particular case when ^ = 0, /cos B dx = x cos B -{- C.
This transformation applies in the special case of products of
the form
cos"* a; sin" a;,
where m and n are both positive integers. For this product may
be written
cos'"aj cos"
(f-)'
ftnd, applying the preceding process, we are led to a sum of sines and
cosines of multiples of the angle, each term of which is immediately
integrable.
As an example let us try to calculate the area of the curve
(1)'^©'
which we may suppose given in the parametric form x = a cos*^,
y = 6 B\n*$, where varies from to 27r for the whole curve. The
formula for the area of a closed curve,
A = -| xdy — ydx,
A=J -^-sin^dcos^drfd.
But we have the formula
(sin e cos ^« = 1 8in«2d = ^ (^ - ^^^ ^^) •
Heooe the area of the given curve is
A-^f^L sin4d"|"'
Sirah
V,§117J TRANSCENDENTAL FUNCTIONS UM
It is now easy to deduce the following formulae *.
I Bin X ox— I 2 *** "2 4 — ^ ^»
/. , . rSsinx — 8in3ar , 3ooez . ooe3a;
fcos'xdx- fl±^^^dx „ ^ I «^° 2g
/- _ r 3 COS r 4- COS 3x , 3 sin X . sin 3x _
C08*xdx=J ^^ dx =__ + __ + c,
/. , r3-|-4cos2x + co8 4x , 3a; . sin2x . sinix . ^
A general law may be noticed in thesr foMini];»-. I h,' ii.t.-n!-
F(x) = J sh\''xdx and <I>(a') = T cos"x^/x have th«' jM-rnxi L'tt
when n is odd. On the other hand, when n is even, theee integrals
increase by a positive constant when x increases by 27r. It is en-
dent a priori that these statements hold in general. For we hare
J pin ptw + s
I sin^xrfxH- I sin'xdlz,
Jtw
or
' sin-o-rfx-f / sin"xrfx = F(x)4- / sin'x(/T.
since sin x has the period 27r. If n is even, it is evident that the
integral j'^'sin^'x rfx is a positive quantity. If n is odd, the same
integral vanishes, since sin (x 4- tt) = — sin x.
Note. On account of the great variety of trausformations appli-
cable to trigonometric functions it is often convenient to iDtrodtiM
them in the calculation of other integrals. Consider, for example,
the integral f 11/(1 -{■ x^)^]dx. Setting a;«tan^, this integral
becomes / cos <(> d<t> = sin <f> -{- C. Henoe, returning to the variable %
/;
^ ^ +c,
(1 4- a-«)« VlTl?
which is the result already found in % 105.
S44 INDEFINITE INTEGRALS [V, §118
118. The integral /R(x)e^dx. Let us now consider an integral
of the form j lt{x)e'^dx, where R{x) is a rational function of x.
Let u« suppoee the function R{x) broken up, as we have done
times, into a sum of the form
where E{x), A^y .4,, .-, A^, J^i, •••, ^p are polynomials, and Z, is
prime to its derivative. The given integral is then equal to the
sum of the integral f E(x)e^''dx, which we learned to integrate in
f 85 by a suite of integrations by parts, and a number of integrals
of the form
"A e'^'dx
i-
There exists a reduction formula for the case when n is greater
than unity. For, since X is prime to its derivative, we can determine
two polynomials X and /* which satisfy the identity ^ = XZ + ^Z'.
Hence we have
and an integration by parts gives the formula
j^^"iF = -;r3i]^+;ririj \n-. dx.
Uniting these two formulae, the integral under consideration is
reduced to an integral of the same type, where the exponent n is
reduced by unity. Repeated applications of this process lead to
the integral
Vie"'
dXf
X
where the polynomial B may always be supposed to be prime to
and of less degree than A'. The reduction formula cannot be applied
to this Integral, but if the roots of X be known, it can always be
reduced to a single new type of transcendental function. For
definiteness suppose that all the roots are real. Then the integral
In quasticm can be broken up into several integrals of the form
J x^a
dx,
X — a
V.jnyj TRANSCENDENTAL FUNCTIONS t46
Neglecting a constant factor, the substitutions s m a 4- y/«, uwm^
enable us to writ., tins integral in either of the fMlV»witjg forms:
!'-?■ /iS
u
The latter integral / [1/log u'\du is a transoendental fuDOtion whidi
is called the integral logarithm.
119. Miscellaneous integrals. Let us consider an integral of the form
f'
where / is an integral function of sin x and ooe x. Any tflrm of
this integral is of the form
I e"8in'"xco8"a;rfx,
where m and n aie positive integers. We have seen above that the
product sin*" a; cos^x may be replaced by a sum of sines and cotinet
of multiples of x. Hence it only remains to study the foUowiny
two types :
I e"' cos bxdx, j ef" sin hx dx.
Integrating each of these by parts, we find tne lormulsB
/, , e^'sin^x o, C ^ . . ,
/. , , e^coste , o r^ . .
e*" sin bxdx = "^ 7 / ^ '*'*'* ^ ^'
Hence the values of the integrals under consideration are
/, , «"(aco8*x + *sinte)
c" cos bxdx = — ^ , ,, '»
/
. . , «"(asm&c — dcoete)
ef^ sm bxdx = — ^ TTTi '
Among the integrals which may be reduced to the preceding
types we may iiHMition the following
j /(log x) X- Jx , / A'^^ sin x) c/x ,
j/{x) arc sin X <ix, J A*) •«? tan x Af,
24ti INDEFINITE INTEGRALS [V, Exs.
wh«re / denotes any integral function. In the first two cases we
should take log x or arc sin a: as the new variable. In the last
two we should integrate by parts, taking f{x) dx as the differential
of another polynomial F(x), which would lead to types of integrals
already considered.
EXERCISES
1. E?alaate the indefinite integrals of each of the following functions :
1 1 z* - g» - 3a;g - a; 1 + vT+x
(«• + !)«' X(X» + 1)«' (X2+1)8 ' \-y/i '
1 l+vTTx
1 + X + VH- x« i-^J/TTx' V« + Vx + 1+ Va;(x + 1)' cos^x'
N
2. Find the area of the loop of the folium of Descartes :
x» + y8 - Zaxy = 0.
8. Eyaluate the integral /y dx, where x and y satisfy one of the following
IdenUties:
(af«-a«)«-ay2(2y^3a) = o, y«(a-x) = x8, y {x^ + y^) = a {y^ - x'^) .
4. Derive the formulaB
r8in-ixcos(n+l)xdx= ^^nnxcosnx
J n '
/.in-ix8in(n + l)x(te= "^°"^J^"^ + c,
rcos-»xcos(n+l)xdx= 2^!!^f!?^^^
/co«-ix8ln(n + l)xdx = - ^^^:i£^^!i2^ + C.
5. EvaluatA each of the following pseudo-elliptic integrals :
r (n-x»)dx r (i-xa)dx
J (i-a!«)vr+^* J (n-x2)vrT^'
e. RMlooe the following integrals to elliptic integrals :
B(«)dx
/
Va(l + «•) + te(i + x«) + cxa(l + x2) -f dx«*
/. ^(«)dx
Va(l + a^) + 6«*(l + x*) + cx*'
Jl(s) draotM a ratioiua funcUon.
v.Exi.] exkkcisf:s 047
7*. Let a, 5, c, d be the roou oi an equauon of tbe fooith degrte t\x) ^ a
Then there exist three involutory relatioiw of tha form
whioh hiterchange the roots by pain. If the rational fanetloD /fz) rritltHM Iha
identity
^'>^x/(-^7^;)-.
the integral f[/{x)/VP{x)]dx is peeudo-ellipUc (Be« BuU$Un de la SociMt
maiique, Vol. XV, p. 106).
8. The rectification of a curve of tbe type y = Azf^ leads to an imeyral of
a binomial differential. Discmui the cases of integrablUty.
9. If a > 1, show that
I
X
1 {a-x)VT^^ Vcfi^
Hence deduce the formula
X
^^ x^*dx ^ 1.8.6.»»(2it-l)
1 Vl-«* 2.4.6..2n
10. Ji AC - B^>0, show that
X
■*^* dx 1.8.6. »(2n- 3) .A--*
(^x2 4. 2Bx + C)" 2. 4. 6. -.(211 -2) (^C-B«)"'^l
[Apply the reduction formula of § 104.]
11. Evaluate the definite Integral
8in> zdx
X
^ l + 2aco«z + a*
12. Derive the following formulte ;
J_i VI - 2ax + a« VI - 2^« + ^ V^ \\-Vafi/
X
* (l-az)(l -^z)dx M t-afi
Li (l-2a« + <r«)(l-2/J« + ^Vrn?"l l-<*^*
IS*. Derive the formula
X
where m and n are positive integers (m<i»). [Break ap the
partial fractions.]
248 INDEFINITE INTEGRALS [V, Exs
14. Flom the prooeding exercise deduce the formola
X
*''^Ll^ = -!L., o<a<l,
1 + X Bin a*
1ft. Setting Ip^^ s: f t' {t + lydt, deduce the following reduction formulse :
(P - 1)/-P.« = <« + M« + 1)'-'' - (2 + g - j))I-p + i,<„
aod two anAlogouB formulse for reducing the exponent q.
IB. Derive formulffi of redaction for the integrals
■ J V-^x« + 25x + c' "* J (X - a)*" V^x2 + 25« + C *
17*. Derive a reduction formula for the integral
r' x"dx
dedace a formula analogous to that of Wallis for the definite integral
X
' dx
la Hm the definite integral
X
dx
1 + «* sin^x
a floite value f
19. Show that the area of a sector of an ellipse bounded by the focal axis
and a radius vector through the focus is
f r do,
2 Jo (l+ccosa,)5
I p danotM the parameter b^/a and c the eccentricity. Applying the gen-
eral method, make the substitutions tan w/2 = <, i = m V(1 + e)/(l - e) succes-
ilfelj, and show that the area in question is
A = a6 ( arc tan u - e — - — J .
\ 1 + uV
Aleo ihow that this expression may be written in the form
A = _(0--esin0),
# le the ecoentrio anomaly. See p. 406.
». Find the carves for which the distance NT, or the area of the triangle
mttT, M ooaetaat (Fig. 3, p. 31). Construct the two branches of the curve.
f Licence, Paris, 1880; Toulouse, 1882.]
V, ExMJ EXERCISES 849
21* Setting
derive the recurrent formula ^
ox
From tbifi deduce the forniuln
where rTsp , Vtp, U-ip^i, Vtp + 1 are polynomial! with intagral o o Mfciwte , tad
where I/^p and U^p + 1 contain no odd powers of z. It !■ rHMlilj thown that
these formula) hold when n = 1, and the general caae followa from th« ahof*
recurrent formula.
The formula for A-j^, enables us to show that n* it inooDiDMiaorahl*. Dor If
we assume that n'^/i - h/a, and then replace x by %/% in Atp^ w» obcain ft
relation of the form
Hj = aPx/^— ^^^^ r\l-*»)«''oot^d«,
\a2.4.0...4p Jo a
where //i is an integer. Such an equation, however, is impoMiUe, for the ilghlF
naud side approaches zero as "p increases indefinitely.
CHAPTER VI
DOUBLE INTEGRALS
L DOUBLE INTEGRALS METHODS OF EVALUATION
GREEN'S THEOREM
120. Continuous functions of two variables. Let z = f(x, y) be a
function of tlie two independent variables x and y which is contin-
nous inside a region A of the plane which is bounded by a closed
contour C, and also upon the contour itself. A number of proposi-
tions analogous to those proved in § 70 for a continuous function
of a single variable can be shown to hold for this function. For
instance, fftven any positive number e, the region A can be divided into
tuhreffums in such a way that the difference between the values of z at
any two points (x, y), (x', y') in the same subregion is less than c.
We shall always proceed by means of successive subdivisions as
follows : Suppose the region A divided into subregions by drawing
parallels to the two axes at equal dis-
tances S from each other. The corre-
sponding subdivisions of A are either
squares of side 8 lying entirely inside C,
or else portions of squares bounded in
part by an arc of C. Then, if the prop-
osition were untrue for the whole region
A, it would also be untrue for at least
one of the subdivisions, say Jj. Sub-
dividing the subregion Ai in the same
manner and continuing the process indefinitely, we would obtain a
sequence of squares or portions of squaies A, A^, .••, a^, ••, for
which the proposition would be untrue. The region A^ lies between
the two lines x = a^ and x = i,, which are parallel to the y axis,
aod the two lines y = c., y = d^, which are parallel to the x axis.
At n increaees indefinitely a, and b^ approach a common limit X,
ftDd ^ and <^ approach a common limit ^, for the numbers a„,
for example, never decrease and always remain less than a fi*ed
Dumber, it follows that all the points of A^ approach a limiting
260
r
1 1 1 \jrmk\ 1
--2^---"^-
' j.
7
z
Flo. 23
VI,§iJi)J INTRODUCTION GHEEN'8 TUJBOREM 2BI
point (X, fx) which lies within or upon the oootoar C. The i«it of
the reasoning is similar to that in § 70 ; if the theorom italad w«c«
untrue, the function /(x, y) could be shown to be diecontiniMMie ai
the point (K, fi), which is contrary to hypotheeie.
Corollary. Suppose that the parallel lines have been rhnem
so near together that the difference of any two values of < in any
one subregion is less than c/2, and let i; be the distance between
the successive parallels. Let (x, y) and {x'y yO be two points inside
or upon the contour C, the distance between which is less than ^,
These two points will lie either in the same subregion or else in
two different subregions which have one vertex in Aft^tT)^ in
either case the absolute value of the difference
/(^, y) -A'', y")
cannot exceed 2€/2 = e. Hence, yiven any positive number «, anofhrr
positive Jiumber rj can be found such that
\/(x, y)-f(x\ y')|<c
whenever the distance between the two points (x, y) and (x\ y*), wkitk
lie in A or on the contour C, is less than rf. In other words, any funo*
tion which is continuous in A and on its boundary C is im(/brai/y
co?itinuous.
From the preceding theorem it can be shown, as in f 70, that
function which is continuous in A (inclusive of its boundary) is m
sarily Jinite in A. If M be the upper limit and m the lower limit of
the function in A, the difference Af — m is called the oteillation. The
method of successive subdivisions also enables us to show that the
function actually attains each of the values m and M at least once
inside or upon the contour C. Let a be a point for which saw
and b a point for which z — 3/, and let us join a and 6 by a broken
line which lies entirely inside C. As the point (jr, y) describes this
line, z is a continuous function of the distance of the point («, y)
from the point a. Hence z assumes every value fi between m and
M at least once upon this line ($ 70). Since a and b can be joined
by an infinite number of different broken lines, it follows that the
function f(x, y) assumes every value between m and if at an infinite
number of points which lie inside of C
A finite region A of the plane is said to be less than / in all its
diihensions if a circle of radius / can be found which entirely
encloses .1 . A variable region of the plane is said to be infinitesimal
25i 1X>UBLE INTEGRALS [VI, §121
in all its dimensions if a circle whose radius is arbitrarily preas-
•igned can be found which eventually contains the region entirely
within it For example, a square whose side approaches zero or an
ellipse both of whose axes approach zero is infinitesimal in all its
dimensions. On the other hand, a rectangle of which only one side
approsches zero or an ellipse only one of whose axes approaches zero
is not infinitesimal in all its dimensions.
121. Doable integrals. Let the region A of the plane be divided
into subregions aj, a,, • • •, a^ in any manner, and let a>, be the area of
the subregiou a<, and A/,- and rrii the limits of f(x, y) in a.-. Consider
the two sums
= ^^iM^, «=X
of which has a definite value for any particular subdivision
of A, None of the sums *!?-are less than wli,* where fi is the area of
the region A of the plane, and where m is the lower limit of /(ic, y)
in the region A ; hence these sums have a lower limit /. Likewise,
none of the sums s ^xq greater than iV/12, where M is the upper limit
ofJXxy y) in the region A ; hence these sums have an upper limit /'.
Moreover it can be shown, as in § 71, that any of the sums »S is
greater than or equal to any one of the sums s ; hence it follows
that
If the function /[x, y) is continuous, the sums S and s approach
a oommon limit as each of the subregions approaches zero in all its
dimensions. For, suppose that i; is a positive number such that the
oscillation of the function is less than « in any portion of A which
is less in all its dimensions than -q. If each of the subregions %,
Oti * "I a. be less in all its dimensions than -q, each of the differences
Jfi — iHi will be less than c, and hence the difference S — s will be
lati than tO, where O denotes the total area of A. But we have
S'-M^s-i + i-r + r-s,
where none of the quantities S — I, I - i\ i' — s can be negative.
Haooe, in particular, / — /'<cO; and since c is an arbitrary posi-
tiTe number, it follows that / = /'. Moreover each of the numbers
S — / and / — # oan be made less than any preassigned number by
•"/t». If) ba oocutant *. If = m = IT, = m, = *, and 5 = «= mO= ifO.-
TtkAM*.
VI, §121] INTRODUCTION GKKKN'S TH£ORKM S6I
a proper choice of c Hence the Bums S and s Lave a ftftmnion limit
/, which is called the double integral of the function /(z, y)
over the region A. It is denoted by the symbol
y)dxd,j^
and the region A is called t\\e field of integration.
If (^.» Vi) ^ ^y P^"^* inside or on the boundaiy of the ftib-
region a„ it is evident that the sum 2/(^o 17,) ». lies between the two
sums S and 8 or is equal to one of tliein. It therefore alto
approaches the double integral as its limit whatever be the method
of choice of the point (^„ i;<).
The first theorem of the mean may be extended without difficulty
to double integrals. Let /(ar, ^) be a function which is continuous
in .4, and let </>(jr, y) be another function which is continuous and
which has the same sign throughout A, For definiteness we shall
suppose that <^(ar, y) is positive in A. If M and m are the limits of
/(x, y) in A, it is evident thiat*
Adding all these inequalities and passing to the limit, we find the
formula
r / fip^y y) 4>(^^ y)dxdy = fif f 4^(x, y)dxdy,
where fi lies between M and m. Since the function J\Xy y) assumes
the value /x at a point (^, rt) inside of the contour C, we may write
this in the form
(1) jj /(x, y)^{x, y)dxdy =j\i, ^)jj^^^<.'^ y)dxd9,
which constitutes the law of the mean for double integrals. If
<^(j-, y) = 1, for example, the integral on the right> JJdxdjff extended
over the region .1 , is evidently equal to the area Q of that
In this case the formula (1) becomes
(2) jJ Ax, y) iix dy = O/i;^, 1,)
• If /(«, y) is a consunt *, w© «hAll h»re If « m «*, aad
equations. The following formula holds, bofrever, with |i ■ *. — Teaw.
254 DOUBLE INTEGRALS [VI, §122
122. Volume. To the analytic notion of a double integral corre-
sponds the important geometric notion of volume. Let fix, y) be
a function which is continuous inside and upon a closed contour C.
We shall further suppose for definiteness that this function is posi-
tifO. Let S be the portion of the surface represented by the equa-
tion « =A*"» y) which is bounded by a curve T whose projection
upon the xy plane is the contour C. We shall denote by E the por-
tion of space bounded by the xy plane, the surface S, and the cylinder
whose right section is C. The region A of the xy plane which is
bounded by the contour C being subdivided in any manner, let a,- be
one of the subregions bounded by a contour c,-, and w,- the area of
this subregion. The cylinder whose right section is the curve c^ cuts
out of the surface 5 a portion s^ bounded by a curve y.- . Let p^ and
P< be the points of «,• whose distances from the xy plane are a mini-
mum and a maximum, respectively. If planes be drawn through
these two points parallel to the xy plane, two right cylinders are
obtained which have the same base co^, and whose altitudes are the
limits 3/,- and m,- of the function /(x, y) inside the contour c.., respec-
tively. The volumes K,- and v\ of these cylinders are, respectively,
^iMi and (i),mf.* The sums S and s considered above therefore repre-
sent, respectively, the sums Sr^ and 2*^,- of these two types of cylin-
ders. We shall call the common limit of these two sums the volume
of the portion E of space. It may be noted, as was done in the case
of area (§ 78), that this definition agrees with the ordinary concep-
tion of what is meant by volume.
If the surface S lies partly beneath the xy plane, the double integral
will still represent a volume if we agree to attach the sign — to the
▼olomes of portions of space below the xy plane. It appears then th^t
every double integral represents an algebraic sum of volumes, just as
a simple integral represents an algebraic sum of areas. The limits of
integration in the case of a simple integral are replaced in the case of a
double integral by the contour which encloses the field of integration.
183. Evaluation of double Integrals. The evaluation of a double
integral can be reduced to the successive evaluations of two simple
integrals. Let us first consider the case where the field of integration
•Bj the voftinM q^ a right eylindMt we shall understand the limit approached hy
tka volama of a right prism of the same heiglit, whose base is a polygon inscribed in
A rifkt itetlOB of the cylinder, as each of the sides of this polygon approaches zero.
r^fcto diiallloB la nol aaoaasary for the argnment, but is useful in sliowing that the
of volttme in general agrees with our ordinary conceptions. — Trans.]
VI, § 123]
IXTRODL'CTIOV r.RFFVS T)fBOR£M
8A6
is a rectangle H bouuded by Uiu straight Udm «hc^ * * X,
y = yo, y = Y, where x^<X and y^O'. SuppoM ihb reeUoflt
to be subdivided by parallels to the two axes r a x^, y ■" y*
(i = 1, 2, , 7t ; /: = 1, 2, • •, m). The area of Uie small reotanfto
72^. bounded by the lines x = x<_„ ac = x^, y = v» _ ,. y s y^ ij
Hence the double integral is the limit of the sum
(;<)
•">• = %%AU, v,t)(^ - - ,)(y. - y.-,).
D
_tf«
Vt
u
(7
m
ft
V9
A
a
•o^
ha
Pw. 24
where (^,4., r^a) is any point
inside or upon one of tlur
sides of Ri^.
"We shall employ the in de-
termination of the points
iiiki Vik) ^^ order to simplify
the calculation. Let us re-
mark first of all that if f(x)
is a continuous function in
the interval (a, b), and if the interval (a, b) be subdivided iu any
manner, a value |, can be found in each subinterval (x^ ,. r^^ §urh
that
(4) j[/(x)^=/(«(aH-«)+/l^O(*t-xO+--.+/(«(*-«^-0.
For we need merely apply the law of the mean for int«graU to each of
the subintervals (a, x,), (xj, «,), • • •, («,_„ b) to find these Tmluea of t •
Now the portion of the sum S which arises from the row of reo-
tangles between the lines x = Xi_| and x = x^ is
(Xi - a^.-i)[/(^.n ViiXyi - Vo) H-/(6f , ViiXsft - yi) + • • •
Let us take ^., = ^,., = • • • = ^<. = x,.„ and then choose i^u %,, •••
in such a way that the sum ~~
/(a:<-„ i7n)(yi - yo) +/[^<-i. %f)(yi - yO + • •
is equal to the integral XV(^<-i» y)<'y» where the integral i« to be
evaluated under the assumption that x<., la a ooostant If we pro-
ceed in the same way for each of the rows of reotanglee bounded by
two consecutive parallels to the y axis, we finally find the eqoatioo
(5) 5 = *(Xo)(xi-Xo) + *(a',)(x,-x.) + ... + *(«,.0(«i-«,.0 + ---»
X
256 DOUBLE INTEGRALS [VI, §123
where we have set for brevity
4<x)= ff(x,y)dy,
Thia function <l>(a:), defined by a definite integral, where x is con-
■idered as a parameter, is a continuous function of x. As all the
intenrals «< — «(_! approach zero, the formula (5) shows that S
i^proaches the definite integral
^(x)dx.
Hence the double integral in question is given by the formula
W rr /(^' y)dxdy= f dx ff(x, y)dy.
In other words, in order to evaluate the double integral, the function
JXZf y) should first he integrated between the limits y^ and F, regard-
ing X as a constant and y as a variable; and then the resulting func-
tion^ which is a function of x alone, should he integrated again hetween
the limits x^ and X.
If we proceed in the reverse order, i.e. first evaluate the portion
of S which comes from a row of rectangles which lie between two
cooaecutive parallels to the x axis, we find the analogous formula
/ I A^i y)dxdy = I dy I f(x, y)dx.
A comparison of these two formulae gives the new formula
' dx I fix, y)dy=\ dy f(x, y)dx,
which is called the formula for integration under the integral sign.
An MMntial presupposition in the proof is that the limits Xq, X, y^, Y
are ooutants, and tliat the function f(x, y) is continuous throughout
the field of integration.
I^t z = xy/a. Then the general formula gives
VhiiTsq INTRODUCTION 0RESN*8 THEOBUC 167
In general, if the function /(x, y) is the product of a fimofeioii ol m
alone by a function of y alone, we shall hare
The two integrals on the right aie absolutely independent of eaeh
other.
Franklin * has deduced from this remark a Tery cimple demoiiKiBtioo ol e«r-
tain interesting theorems of Tchebycheff. Let ^(x) and f (x) be two fa
which are continuous in an intenral (a, b), where a < 6. Then the double I
ffM^) - *(y)] M"^) - nv)]dxdv
extended over the square boutidod by the Unee z = a,z = 6, y = a,yB6is
to the difference
2(6- a)j^ 0(x)^(x)(te - 2f\(x)dx x j\{z)dz.
Bat all the elements of the above double integral have the mido sigB if the two
tunctions 0(x) and ^(x) always increaee or decrease aimaltaneooaly, or If om of
them always increases when the other decreasee. In the flntcaae tho twofono-
tions <t>{x)-4>{y) and ^(x) - ^(y) always have the Mine sign, whenas ib^ haw
opposite signs in the second case. Hence we shall have
(6 - a)j\{x)i^(x)dx >f\(x)dx xf%{x)dx
whenever the two functions 0(x) and ^(x) both increase or both decrease throQfh-
out the interval (a, b). On the other hand, we shall have
(6 - a)fy{r)rl^{x)dx < £i^{x)dx x^V(«)*t
whenever one of the functions increases and the
interval.
The sign of the double integral is also definitely detemUaad tn oan ^) m f{z)^
for then the integrand becomes a perfect sqoare. In this case w« ahaU have
(6 - a)f\i.{x)ydz > [ j]Vx)dr]',
whatever be the function 0(x), where the sign of equality can hold only wh«i
0(x) is a constant.
The solution of an interesting problem of the oaleolas of variatioas bmj be
deduced from this result. Let P and Q be two flzod points in a plaae whose
coordinates are (a, A) and (6, B), respectively. Let y =/[*) ^ the eqjaatloB of
any curve joining these two points, where /(z), together with Hi flitt dwlt ell fa
•American Journal qf MathBmatim^ VoL VII, p. 77.
258
DOUBLE INTEGRALS
[VI, § 124
/'(x), if suppoied to be continuous in the interval (a, b). The problem is to
flDd that one of the curves y=f{x) for which the integral J^*y' 2^3; ig a
minimain. But by the formula just found, replacing <p{x) by y" and noting
that /{a) = A and /{h) = B by hypothesis, we have
(6
a)fy'^dx^{B-A)^.
V
A
/
^
B'
1
^A
B
"^
k
^
^
1
!
p \ !
"^
•
X
•-1
«i :
C a
Fio. 25
TTie minimum value of the integral is therefore {B - A)^/{h - a), and that value
It actually a»umed when y' is a constant, i.e. when the curve joining the two
Used poinu reduces to the straight line PQ.
124. Let us now pass to the case where the field of integration is
bounded by a contour of any form whatever. We shall first suppose
that this contour is met in at most two points by any parallel to the
y axis. We may then suppose that it is composed of two straight
lines X — a and x = h (a<b)
and two arcs of curves APE
and A'QB' whose equations are
Yi = <fii(x) and Y^ = <t>2(x), re-
spectively, where the functions
<t>i and <^2 are continuous be-
tween a and b. It may happen
that the points A and A' coin-
cide, or that B and B' coin-
cide, or both. This occurs, for
instance, if the contour is a convex curve like an ellipse. Let us
again subdivide the field of integration R by means of parallels to
the axes. Then we shall have two classes of subregions : regular if
they are rectangles which lie wholly within the contour, irregular
if they are portions of rectangles bounded in part by arcs of the
contour. Then it remains to find the limit of the sum
5 = 2/(^,^)0,,
where « is the area of any one of the subregions and (f, 7;) is a point
in that subregion.
Let us first evaluate the portion of S which arises from the row
of subregions between the consecutive parallels a; = a;,_i, x = «,.
These subregions will consist of several regular ones, beginning
with a Yertex whose ordinate is y' > Y^ and going to a vertex whose
ordinate is y" < K,, and several irregular ones. Choosing a suitable
point (t 17) in each rectangle, it is clear, as above, that the portion
of S which comes from these regular rectangles may be written in
the form
VI, §124] INTRODUCTION GREEN'S THEOREM 269
Suppose that the o8(;illation of each of the functioiui ^t(*) »o<i ^iC*)
in each of the intervals (ar^_), x^ is less than 3, and that each of the
differences y^ — y^_, is also less than & Then it is easily seen that
the total area of the irregular subregions between x « x,., and x ss &
is less than 48(x, —x^_{)y and that the portion of S which artset
from these regions is less than 4//£(a:| •> ob^.,) in absolute value,
where // is the upper limit of the absolute value of /|[z, y) in the
whole field of integration. On the other hand> we have
A^i^u y)dy=^\ /(ar,_„ y)dy{' \m+ I .,
and since | K, — if\ and | Y^ — y"| are each less than 2S, we may write
f A^i-iy y)dy =J^ f{x,,u y)dy + iHxi. |X«1.
The portion of S which arises from the row oi subregions under
consideration may therefore be written in the form
(^. - ^.-i) [J^ /(Xi.i, y)dy + 8W,«J,
where d, lies between — 1 and -\- 1. The sum 8^SS^«(Xj — X|.|) is
less than ^Hh(b — a) in absolute value, and approaches sero with ^
which may be taken as small as we please. The double integral is
therefore the limit of the sum
where
♦(«)= fk^>y)dy'
Hence we have the formula
(7) / / /(^, y;"^ '^y = f dx f f(x, y)dy.
In the first integration x is to be regarded as a oonitani, bat
the limits Yi and }\ are themselves funotiona of x and not
constants.
260 DOUBLE INTEGRALS [VI, §124
Ji^mplo Let iw try to evaluate the double integral of the function xy/a
m inlerior of a quarter circle bounded by the axes and the circumference
x2 + ya_ij2 = o.
Tl>e UmlU forx are and i2, and if « ifl constant, y may vary from to VJB* - x^.
theintagnlis
^-jr>.Mo^--=X
*x(B^-x^)^_
a " J, 2a L Jo J^ 2a
The Tilue of the latter integral is easily shown to be R*/Sa.
When the field of integration is bounded by a contour of any form
whatever, it may be divided into several parts in such a way that
the boundary of each part is met in at most two points by a parallel
to the y axis. We might also divide it into parts in such a way that
the boundary of each part would be met in at most two points by
any line parallel to the x axis, and begin by integrating with respect
to X. Let us consider, for example, a convex closed curve which lies
inside the rectangle formed by the lines x = a^ x = bj y = c, y = d,
upon which lie the four points AjB, C, D, respectively, for which x
or y is a minimum or a maximum.* Let y^ = 4>i(oc) and 3/2 = <^2(^)
be the equations of the two arcs ACB and ADB, respectively, and
let X, s= ^i(y) and «, = ^tO/) he the equations of the two arcs CAD
and CBDf respectively. The functions <t>i(x) and <f>2(x) are continu-
OQi between a and i, and ipi (y) and \j/2 (y) are continuous between c
and (L The double integral of a function /(a;, y), which is continuous
inside this contour, may be evaluated in two ways. Equating the
ralues found, we obtain the formula
dx f(x,y)dy= / dy f(x, y)dx.
It is clear that the limits are entirely different in the two integrals.
Erery convex closed contour leads to a formula of this sort. For
•lample, taking the triangle bounded by the lines y = 0, cc = a,
y ia X as the field of integration, we obtain the following formula,
which is due to Lejeune Dirichlet :
jf dx^Jix, y)dy=j^dyjf(x, y)dx.
*TlM reader U advised to draw tlie figure.
Vi,5125j INTRODUCTION GRKEN'S TBEOEEM SSI
125. Analogies to simple iatsfrala. The iatflgnl f'/{t^M^ Tini^ilind M %
function of z, has the deriTAtiTe /(z). There eziete aa ^^•^'r^ arm Ibeons te
double integrals. Let /(z, y) be a funotion which to oootiaoooi taMlde a m»»
tangle bounded by the straight lines x^a^x^ A,y ^b,ym B,{(^ <A^h<B^
The double integral of /(z, y) extended over a rectaofia >*i*n fMitil by tiM ItaMS
x^a,x = X,v = b,y- y,(a<J:<^, 6<r<J3),toafiiii0iloBof IkaMfli^
nates X and Y of the variable comer, that is,
F(X.r)=^'*dzj[7(z,y)dy.
Setting 4>(z) =^ /(z, v)dy, a first differentiation with respect to X ^^m
A second differentiation with respect to Y leads to the fomnila
*«' ^-|^=^'•^>•
The most general function u{X, F) which tatliflM tba eqnatkm (0) to tfl-
dently obtained by adding to F(.Y, Y) a funotion % wboaa nooad <torivatlf«
d'^z/dXiiY is zero. It is therefore of the form
(10) M(X, F) = fdz r7(z, y)dy + ^(X) + ^(F),
where 0(X) and ^( F) are two arbitrary functions (see 1 88). The two aitltraiy
functions may be determined in such a way that «(X, F) red o cea to a fivta
function r(F) when A" = a, and to another given fnaetion Cr(X) ahl Ts li
Setting AT = a and then F = & in the preceding e(|Qattoo, we obtain tha two
conditions
V{Y) = 0(a) + f (F), Cr(X) = *(X> + ^^(6).
whence we find
^(F) = V(J) - 0(a) , ^(6) = 7(6) - ^(a) . ♦(X) = ir(X) - r(k) + #(«),
and the formula (10) takes the form
(11) t^(X. F) = j^^dz j^7(z, y)dy + ir(X) + F(F) - 7(6) .
Conversely, if, by any means whatever, a fnnetion ii<X, F) baa bai
which satisfies the equation (0), it is easy to show bj methods ilnUar to lAt
above that the value of the double integral to given Uy tba formula
(12) /'iix/ M V)dv = u(X. F) - u{X, 6) - u{a, Y) + «(a, 6).
This formula i.s analogous to the fondamental fononln (6) on
The following formula is in a sense analofoos to ttot foimito for
by parts. Let ^ be a finite region of the plane boondad bj ons or i
262 DOUBLE INTEGRALS [VI, §126
of any form. A function /(x, y) which is continuous in A varies between its
mlnlmam «« and ita maximum V. Imagine the contour lines /(x, y) = v drawn
whU9 9 Uas between »o &Qd F, and suppose that we are able to find the area of
the portkOB of A for which /(x, y) lies between vq and v. This area is a func-
tion /t*) ^tiioh increases with o, and the area between two neighboring contour
llMtle ^(v + Ae) — F{v) = AoF'(o + ^Ao). If this area be divided into infinitesi-
■ftl poitioni by lines joining the two contour lines, a point ({, rj) may be found
In eaeh of tbem such that /((, 17) = v + SAv. Hence the sum of the elements
of the double integral Jf/dx dy which arise from this region is
(r + 0Av) F\v + dAv) At>.
It follows that the double integral is equal to the limit of the sum
2(t> + ^ At)) F\v + ^ Ao) At) ,
that is to say, to the simple integral
J t» F\v) dv = VF{ y)-f FCo) dv .
This method is especially convenient when the field of integration is bounded
by two contour lines
/(x, y) = »o, /(x, y)= V.
For example, consider the double integral J f y/\ + x^ -\-y^dxdy extended over
the interior of the circle x^ -f- y2 _ j. if ^^ set v = Vl + x'-^ + y'^, the field of
integration Lb bounded by the two contour lines v = \ and v = V2, and the
foncUon F{v), which is the area of the circle of radius Vt)^ - 1, is equal to
«(•• - 1). Hence the given double integral has the value
j^ '2>ro«d« = ^(2%^-l). *
The preceding formula is readily extended to the double integral
fff(x,y)<P{x,y)dxdy,
F{v) now denotes the double integral //0(x, y)dxdy extended over that
of the field of integraUon bounded by the contour line v = /(x, y).
^ lae. Green»e theorem. If the function /(a;, y) is the partial deriva-
tive of a known function with respect to either x or y, one of the
tntagrations may be performed at once, leaving only one indicated
mtogration. This very simple remark leads to a very important
formula which is known as Green's theorem.
m wT TTl*"!!!*^*"* **' '***■ "•"***** are to be found in a memoir by Catalan
(/OMTMI 4s JJawHiU, 1st series, Vol. IV, p. 233) .
Vl,Jii(ij IMKODUCTION GKKEN'S THKjOHEU
Let us consider Brst a double integral ffdP/e^dxdy exiemM
over a region of the plane bounded by a contour C, which it met
in at most two points by any line parallel to the y axis (see Fig. IS^
p. 188).
Let A and B be the points of C at which x is a minimum ftiH a
maximum, respectively. A parallel to the y axil between Aa and
Hb meets C in two points m^ and m, whose ordinates are yx ^^^ y»
respectively. Then the double integral after integration with reapeei
to y may be written
But the two integrals ^ P{Xy yx)dx and /V(x, y^dx are line
integrals taken along the arcs ArriiR and Am^B, respectlTely ;
the preceding formula may be written in the form
(13) fjf^a.ay = -jja..
where the line integral is to be taken along the contour C in the
direction indicated by the arrows, that is to say in the positive
sense, if the axes are chosen as in the figure. In order to extaod
the formula to an area bounded by any contour we should proceed
as above (§ 94), dividing the given region into several parts for eaeb
of which the preceding conditions are satisfied, and applying the for-
mula to each of them. In a similar manner the following analogous
form is easily derived :
(14) //g</W,=£«iy,
where the line integral is always taken in the same sense. Sob-
tracting the equations (13) and (14), we find the formula
(15) //^■'^'^=//(S-'4)
d^dy.
where the double integral is extended over the region bounded by C,
This is Green's formula ; ite applications are very important Just
now we shall merely point out that the substitution Q = x and
p = -y gives the formula obtained above (§ W) for the area of a
closed curve as a line integral.
264 DOUBLE INTEGRALS CVI,S127
IL CHANGE OF VARIABLES AREA OF A SURFACE
In the evaluation of double integrals we have supposed up to the
present that the field of integration was subdivided into infinitesimal
reetangles by parallels to the two coordinate axes. We are now going
to suppose Uie field of integration subdivided by any two systems of
ourres whatever.
127. Preliminary formula. Let u and v be the coordinates of a point
with respect to a set of rectangular axes in a plane, x and y the coor-
dinates of another point with respect to a similarly chosen set of
rectangular axes in that or in some other plane. The formulae
(16) x=f(uyv), y = 4>(u,v)
establish a certain correspondence between the points of the two
planes. We shall suppose 1) that the f unctions /(w, v) and <^(w, v),
together with their first pai-tial derivatives, are continuous for all
points (u, v) of the uv plane which lie within or on the boundary of
a region Ai bounded by a contour Ci ; 2) that the equations (16)
transform the region A^ of the uv plane into a region A of the
xy plane bounded by a contour C, and that a one-to-one correspond-
ence exists between the two regions and between the two contours
in such a way that one and only one point of ^ j corresponds to any
point of i4 ; 3) that the functional determinant A = D(f, <f>)/D(u, v)
does not change sign inside of Ci, though it may vanish at certain
points oi Ai.
Two cases may arise. When the point (w, v) describes the con-
tour C, in the positive sense the point {x, y) describes the contour C
either in the positive or else in the negative sense without ever
reversing the sense of its motion. We shall say that the corre-
spondence is direct or inverse, respectively, in the two cases.
The area O of the region A is given by the line integral
Jtc
xdy
(C)
taken along the contour C in the positive sense. In teims of the
new variables u and v defined by (16) this becomes
wbare the new integral is to be taken along the contour C^ in the
positive §miM^, and where the sign + or the sign - should be taken
VI, J 127]
CHANGE OF YAiUAULBS
206
according as the correspondence is direct or inverse. Applying
Green's theorem to the new integral with x » v, o ai y, /> ^fd^/bu^
Q=/a<^/ay, wefind
du
wlience
^-j;-^
or, applying the law of the mean to the (louriie integral,
(17)
0=:±0i
where (^, 17) is a point inside the contoar Ci, and Oi is the area of
the region .li in the uu plane. It is clear that the sign + or the
sign — should be taken according as A itself is positiTc or negative.
Hence the correspondence is direct or inverse OMording as A is positive
or negative.
The formula (17) moreover establishes an analogy between func-
tional determinants and ordinary derivatives. For, suppose that the
region A 1 approaches zero in all its dimensions, all its points approacb-
ing a limiting point {iiy v). Then the region A will do the same, and
the ratio of the two areas O and 0^ approaches as its limit the abso>
lute value of the determinant A. Just as the ordinary derivative is
the limit of the ratio of two linear infinitesimals, the functional
determinant is thus seen to be the limit of the ratio of two infinites-
imal areas. From tliis point of view the formula (17) is the analogoo
of the law of the mean for derivatives.
Bemarki. The hypotheaes which we have made oonoemlng the (
between ^ and ^ 1 are not all Independent Thua, In order that tke
ence should be one-to-one, it is neceaeary that A ■hookl not ehaafi ilga in Iha
region ^1 of the uv plane. For, suppose that A vanishes along a oorva -n which
divides the portion of A\ where A is
positive from the portion where A is
negative. Let us consider a small aro
mi nx of 7i and a small portion of A\
which contains the arc mini- Thla
portion is composed of two regions a\
and a'x which are separated by mini
(Fig. 20).
When the point (u, t) deacribea the fto. tS
region a\, where A is positive, the point
(x, y) describes a region a bounded by a coDtoor MapM, and the two
mi ui pi mi and mnpin are described aimultaneooaly In the poritlve ansa
the point (m, v) describes the region «(, where A Is negative, the polat (s, f)
266 DOUBLE INTEGRALS [VI, §128
ilMTiillim a regUm a' whose contour nmqr is described in the negative sense as
Hi Ml 9i Hi it deaeribed in the poeitive sense. Tlie region a' must therefore
eovw a part of the region a. Hence to any point (x, y) in the common part
arm oorrespond two points in the uv plane which lie on either side of the
line mi»i.
As an ezAinpIe consider the transformation X = x,Y = y^, for which A = 2y.
If the point (z, y) describes a closed region which encloses a segment ab of the
X axis, it is eTident that the point (X, Y) describes two regions both of which
lie above the X axis and both of which are bounded by the same segment AB of
that axis. A sheet of paper folded together along a straight line drawn upon it
fiTSs a clear idea of the nature of the region described by the point {X, Y).
The condition that A should preserve the same sign throughout ^i is not suf-
ficient for one-to-one correspondence. In the example X = x'^ — y^, Y = 2 xy,
the Jacobian A = 4 («* + y*) is always positive. But if (r, d) and {R, w) are the
polar coordinates of the points (x, y) and (X, Y), respectively, the formulae of
transformation may be written in the form R = r^^ u = 2d. As r varies from a
to 6 (a < 6) and $ varies from Oto?r + a(0<a< 7t/2), the point {R, u) describes
a circular ring bounded by two circles of radii a^ and 62. But to every value of
the angle u between and 2a correspond two values of ff, one of which lies
between and a, the other between 7t and it -\- a. The region described by the
point (X, Y) may be realized by forming a circular ring of paper which partially
orerlape itself.
188. Transformation of double integrals. First method. Retaining
the hypotheses made above concerning the regions A and Ai and the
formulae (16), let us consider a function F(x, y) which is continuous
in the region A. To any subdivision of the region A i into subregions
On a^y •y a^ corresponds a subdivision of the region A into sub-
regions a,, a,, • . •, a,. Let w, and o-, be the areas of the two corre-
sponding Bubregrions a< and a,., respectively. Then, by formula (17),
Wf = 0-.
where u^ and v, are the coordinates of some point in the region a,.
To this point (m„ t;<) corresponds a point x, =f(Ui, v,), y, = <^(w., v,)
of the region a<. Hence, setting <I>(w, v) = F[f(u, v), <f>(u, v)], we
may write
<- 1 i -I
whanee, pasting to the limit, we obtain the formula
(.8, fin-, y)^^y -/X/C/C. "). *(«. V)] I 'j^^Uu,..
VI. 5 128]
CHANGE OF \ AKlAbLlv.->
267
Hence to perfomn a transformation in a double imUgral x amd y rh mU
be replaced by their values as functions of ths new variaAlee u mmd 9
and dxdy should be replaced by \^\dude. We bare moo tlraadr
how the new field of integration is determined.
In order to find the limits between which the integrmttons ihould
be performed in the calculation of the new doaUe integral, it ii ia
general unnecessary to construct the contour C, of the new field
of integration .l^ For, let us consider i« and tr m a fjaten of
curvilinear coordinates, and let one of the yariaUet u and v in tha
formulae (16) be kept constant while the other varies. We obtain
in this way two systems of curves u = const, and v » eonst By
the hypotheses made above, one and only one curre of each of iheaa
families passes through any
given point of the region A.
Let us suppose for definite-
ness that a curve of the
family v = const, meets the
contour C in at most two
points Ml and 3/, which cor-
respond to values u^ and w,
of u (wj < ifj), and that each (p
of the (v) curves which meets
the contour C lies between
the two curves v = a and
V = 6 (a<b). In this case
we should integrate first wm,ti
with regard to u, keeping v constant and letting u rary from ti|
to tzj, where xi^ and u^ are in general functions of r, and then inte-
grate this result between the limits a and b.
The double integral is therefore equal to the ezpreation
M
F[/(«, r), ^(«,r)]|Alifii,
A change of variables amounts essentially to a snbdiTiaion of the
field of integration by means of the two SjTStems of curres (m) and (v).
Let 0) be the area of the curvilinear quadrilateral bounded by the
curves («), {u -h du), (v), {v 4- dv), where du and rfr are poaitiTe.
To this quadrilateral corresponds in the uv plane a rectangle whose
sides are du and dv. Then, by formula (17), • = | A(^, iy)| dm «/r, wbove
^ lies between u and u + du, and iy between v and v + dv. The exprea-
sion I A(u, r) | du dv is called the eUmml ^f arem in the tjwUm id
268 DOUBLE INTEGRALS [VI, §129
ooftidinates (u, v). The exact value of a> is w = J | A(w, v)\-\-€ldu dv,
where c approaches zero with du and dv. This infinitesimal may be
na^acted in finding the limit of the sum ^F{x, y) <o, for since A(w, v)
U oontinuous, we may suppose the two (w) curves and the two
(v) oarret taken so close together that each of the e's is less in ab-
Boliite Talue than any preassigned positive number. Hence the abso-
lute value of the sum SZ-X^, y)idudv itself may be made less than
any preassigned positive number.
189. Examples. 1) Polar coordinates. Let us pass from rectangu-
lar to polar coordinates by means of the transformation x = p cos <o,
y = /> sin <u. We obtain all the points of the xy plane as p varies
from zero to -I- oo and o> from zero to 27r. Here A = /» ; hence the
element of area is p dm dp, which is also evident geometrically. Let
us try first to evaluate a double integral extended over a portion of
the plane bounded by an arc AB which intersects a radius vector in
at most one point, and by the two straight lines OA and OB which
make angles wi and wj with the x axis (Fig. 17, p. 189). Let
R = ^(«) be the equation of the arc AB. In the field of integration
m varies from o>i to (i>s and p from zero to R. Hence the double inte-
gral of a function /(x, y) has the value
\ dm\ f{
p COS o), p sin 0)) p dp.
If the arc AB ia a. closed curve enclosing the origin, we should
take the limits <i»i = and 0)3 = 27r. Any field of integration can
be divided into portions of the preceding types. Suppose, for
instance, that the origin lies outside of the contour C of a given
convex closed curve. Let OA and OB be the two tangents from
the origin to this curve, and let R^ =fi(ui) and R^ =/2(<o) be the
equations of the two arcs ANB and A MB, respectively. For a
given value of w between <i>i and w^, p varies from i2, to R^, and
the value of the double integral is
f(p cos o), p sin <a)pdp.
fl) EU^k eoiirdinaUM. Let ui consider a family of confocal conies
VI, $ 130]
CHANGE OF VARIABLKS
SM
where X denolM an arbitrary parameter. Tbroogh urerj point of the
two conies of thia family, — an aiUpie and an hyperbola, — for Iho
r
w
»S
m
OV-^^-^F
has one root X greater than (^, and another posiUte root |i
values of x and y. From HO^ and from the analogont
replaced by m we find
than ^, for any
X la
(20)
VXm
V^(X-c«)(c*-m)
0<^^<^^X.
To avoid ambiguity, we shall consider only the first quadrant in tlie
This region corresponds point for point in a one-to-one mannur to tba
the X/i plane which is bounded by the straight lines
X = c», M = 0. /» = <^.
It is evident from the formulie (20) that when the point (X, f) deaBribei tko
boundary of this region in the direction indicated by the arrowa, the point (x, y)
describes the two axes Ox. and Oy in the aenie i ndicated by the arrowa. Tbe
transformation is therefore inverse, which is verified by calcniating A :
^^ J>(x, y) ^ 1 X-n
D(X, n) 4 VXM(X-r«)(c«-^)*
130. Transformation of double integrala. Second method. We ihall
now derive the general formula (18) by another method which
depends solely upon the rule for calculating a double integral. We
shall retain, however, the hypotheses made aboTe ooooemiDg the
correspondence between the points of the two regions A and Ax*
If the formula is correct for two particular traasformatioos
it is evident that it is also correct for the transformation obtained
by carrying out the two transformations in mooession This fottows
at once from the fundamental property of foBetknal detenninants
(§80)
D{T. y) i>(ar, y) D(u, v)
D{u\ u') D{u, v) D(u\ rO
S70
DOUBLE INTEGRALS
[VI. § 130
Similarly, if the formula holds for several regions A,B,C, ",L,
to which correspond the regions Ay, Bi, Ci, ••-, Li, it also holds for
the region A -^ B -k- C -i \- L. Finally, the formula holds if the
trmnsfonnation is a change of axes :
X = ar^ -f x' COS a - y' sin a, y = yo + x' sin a -\- y' cos a.
Here A = 1, and the equation
F(Xf y) dx dy
If'
■II'
F(x^ 4- x' cos or — y' sin a, y^ + x' sin a + y' cos a) dx' dy'
is ffttisfied, since the two integrals represent the same volume.
We shall proceed to prove the formula for the particular trans-
formation
(21)
x = <^(x', y'), y = y
which carries the region A into a region ^' which is included between
the same parallels to the x axis, y — yo and y =zy^. We shall sup-
pose that just one point of A corresponds to any given point of A ' and
conversely. If a paral-
lel to the X axis meets
the boundary C of the
region A in at most two
points, the same will be
true for the boundary
C" of the region yl'. To
any pair of points itiq
and mi on C whose or-
dinates are each y cor-
respond two points w'o
mjr
frriA /ffi]
s:
Tnp;
Fio. 29
■ad ml of the contour C. But the correspondence may be direct or
tnTsrse. To distinguish the two cases, let us remark that if d<l>/dx' is
positire, x increases with x', and the points mo and m^ and ?wi and
*h' lie as shown in Fig. 29 ; hence the correspondence is direct. On
the other hand, if dif>/dx' is negative, the correspondence is inverse.
Ut us consider the first case, and let Xo, Xj, x^, x| be the abscissse
of the points m,, m,, w;, mj, respectively. Then, applying the for-
mula for change of variable in a simple integral, we find
jf V y)dx =jr F[^(x', y% y'] grfx',
VI, §130] CHANGE OF VABUBLE8 271
where y and y' are treated as oomtonta. A lingle intA^frrAiion girtt
the formula
^ dyj\x, y)dx =f\'f'nH'\ y'), yl Ijrf''.
But the Jacobian A reduces in this case to d^/9x\ aod hence Uie
preceding formula may be written in the form
ff F(x, y)dxdy = ff F[^(x', y*), y'] } A | rfx'rfy'.
This formula can be established in the same '"^nnfw if d^fdx' is
negative, and evidently holds for a region of any form whatever.
In an exactly similar manner it can be shown that the trans-
formation
(22) x = x\ y = ^(x-, yO
leads to the formula
/ / ^\^y y)dxdy=\\ F[x', ^(x\ y')]|Ai<ix'c/y',
J J(A) J J(A')
where the new field of integration A* oonesponds point for point U*
the region A.
Let us now consider the general formulae of transformatioo
(23) «=A«i, yi), y=/i(afi, yi),
where for the sake of simplicity (x, y) and (Z], yi) denote the coor-
dinates of two corresponding points m and My with retp»c t to the
same system of axes. Let A and .-1 1 be the two oorresponding regioot
bounded by contours C and Ci, respectively. Then a third point m\
whose coordinates are given in terms of those of m and Af, by the
relations x' = x^j y' = y, will describe an auxiliary region A\ which
for the moment we shall assume corresponds point for point to each
of the two regions A and A^. The six quantities x, y, x,, yj, x*, y
satisfy the four equations
x==f(xi>yi)y y=/i(«i,yi), «'-*i, y-yi
whence we obtain the relations
(24) a:' = x„ y' =»/i(«„ y,).
which define a transformation of the type (22). From the oqu:ition
y' =fi(x', y,) we find a relation of tho form y, = Tr(x', y') ; hrnce
we may write
(26) x=/(x',yO=^(ar',y'),
272 DOUBLE INTEGRALS [VI, § 131
The given transformation (23) amounts to a combination of the two
tpansformationa (24) and (26), for each of which the general formula
holds. Therefore the same formula holds for the transformation (23) .
Remark. We assumed above that the region described by the
point m' corresponds point for point to each of the regions A and
A I. At least, this can always be brought about. For, let us con-
sider the curves of the region Ai which correspond to the straight
lines parallel to the x axis in ^. If these curves meet a parallel to
the y axis in just one point, it is evident that just one point m' of
A' will correspond to any given point m of A. Hence we need
merely divide the region Ai into parts so small that this condition
is satisfied in each of them. If these curves were parallels to the
y axis, we should begin by making a change of axes.
181. Area of a curved surface. Let 5 be a region of a curved sur-
face free from singular points and bounded by a contour r. Let S
be subdivided in any way whatever, let 5»- be one of the subregions
bounded by a contour y,-, and let m^ be a point of s^. Draw the tan-
gent plane to the surface S at the point w,-, and suppose s^ taken so
small that it is met in at most one point by any perpendicular to
this plane. The contour y< projects into a curve y- upon this plane ;
we shall denote the area of the region of the tangent plane bounded
by yl by o-^. As the number of subdivisions is increased indefinitely
in such a way that each of them is infinitesimal in all its dimensions,
the sum 2<r,- approaches a limit, and this limit is called the area of
the region S of the given surface.
Let the rectangular coordinates a;, y, « of a point of S be given iii
terms of two variable parameters u and v by means of the equations
(26) x=/(m, v), y = <t>(u,v), z = xf;(u,v),
iu such a way that the region S of the surface corresponds point for
point to a region R of the uv plane bounded by a closed contour C.
We shall assume that the functions /, <^, and ^, together with their
first partial derivatives, are continuous in this region. Let R be
subdivided, let r, be one of the subdivisions bounded by a contour c.,
and let «| be the area of r<. To r< corresponds on ^ a subdivision s<
bounded by a contour y<. Let or< be the corresponding area upon the \
tangent plane defined as above, and let us try to find an expression
for the ratio a,/w|.
I^t rto A» yt be the direction cosines of the normal to the surface S
At A point m,(ar<, y<, «^) of «. which corresponds to a point (w., v,)
VI, § 131]
CHANGE OF VARIABLES
m
of r^. Let us take the point m^ aa a new origin, md m the new
the normal at m, and two perpendicular lines m^X and m^K in lh«
tangent plane whose direction cosines with reepect to the old asee ar«
a', p\ y' and a", /3", y", respectively. Let X,Y,Zhb the
of a point on the surface S with respect to the
by the well-known formulae for transformation of infttrttimfctw^ we
shall have
J^ = a'(x-x,) + /9'(y-y,) + y'(s_i^J,
Y = a'\x - X,) + ^"(y - y,) + /'(, - ,J,
Z = a, (X - x,) -f A (y - y,) + y, (s - «,) .
The area o-< is the area of that portion of the XY plane which is
bounded by the closed curve which the point (X^ Y) deecribet, at
the point (w, v) describes the contour c,. Hence, by f 127,
D{X, Y)
D{uU v^
where u\ and v[ are the coordinates of some point inside of r
easy calculation now leads us to the form
+ (^'''"-''V')|^-H(«'r-/»'«.")^.
<r,- — «»^
or, by the well-known relations between the nine direotioo
/)(«;, rO \ "' D(ul, vl) ^ f^ D(uU r{) * ^' ZK-J, O S
Applying the general formula (17), we therefon obtain the eqoatioo
^ D(ji^ Diz^l D^Ua.1
<r.=
where wj and vj are the coordinates of a point of the region r^ in the
uv plane. If this region is very small, the point (iij, rj) is reiy
the point (m^, v<), and we may write
/>(s,x) ^ D(«,x)
Scr^ = 2<i»i
«.
where the absolute value of $ does not ezoeed unilj. Bines the
derivatives of the functions /, ^, and f are oontinuoos in the
274 DOUBLE INTEGRALS [VI, §131
ragion R, we may assume that the regions r.. have been taken so
small that each of the quantities c<, c^, c]' is less than an arbitrarily
pieassigned number 17. Then the supplementary term will certainly
be lees in absolute value than SrfQ, where ft is the area of the
region R. Hence that term approaches zero as the regions s,
(and r<) all approach zero in the manner described above, and the
sum S<r< approaches the double integral
dudv,
where a, ft y are the direction cosines of the normal to the surface S
at the point (k, v).
Let us calculate these direction cosines. The equation of the
tangent plane (§ 39) is
^^ '^ D(u, v)^^^ y^ D{u, v) ^ ^ ^ D(u, v) "'
whence
a P y ±1
D(u, v) D{u,v) D{u, v) \Li>(w, v)J
Choosing the positive sign in the last ratio, we obtain the formula
^ ^(y> ^) , q DJ^^ X) D(x, y)
D{uy v) ^ D(u, v) ^ D{u, v)
\Lz>(w, v)J ^ [_D(u, v)J ^ LD{u, v)J
Tht well-known identity
(ah' - bay + {be' - cb')* -f (ca' - acy
= (a« 4- *• + c'Xa'^ + ^'' + c'^) - (aa' + bb' + cc')«,
which was employed by Lagrange, enables us to write the quantity
the radical in the form EG — F\ where
<^ --s(^ij> F^stt' <^-s(iJ'
the symbol S indicating that a; is to be replaced by y and z succes-
sifelj and the three resulting terms added. It follows that the area
of the faifaee S is given by the double integral
(38) A =Jj y/EG - F* du dv .
VI, $ 132] CHANGE OF VARIAaLBS 876
The functions E, F, and G play an important part in tha Umocj
of surfaces. Squaring Uie expressions for dx^ dy, and dm and adding
the results, we find
(29) d8* = dx*'hdy* + dx*=E du* -f 2Fduuv -r >. dv».
It is clear that these quantities /?, F, and G do not depend upon
the choice of axes, but solely upon the surface S itaelf and the inde-
pendent variables u and v. If the variablea ii and v and the tor-
face S are all real, it is evident that EG — F* must be poiiitiv<».
132. Surface element, i ne expression vEG — F*dudv ia caliea tA$
element of urea of the surface .S* in the system of co5rdinates (m, v).
The precise value of the area of a small portion of the surfa ce bounded
by the curves (w), (« -|- rfu), (v), (v -f dv) is {^EG — F* + t)dud9,
where e approaches zero with du and dv. It is evident, as abore,
that the term c du dv is negligible.
Certain considerations of differential geometry oontirm in is reeuit
For, if the portion of the surface in question be thought of aa a amall
curvilinear parallelogram on the tangent plane to ^ at the point («, «),
its area will be equal, approximately, to the product of the lengthi
of its sides times the sine of the angle between the two cunrea («)
and (v). If we further replace the increment of arc by the differ-
ential dSf the lengths of the sides, by formula (29), are -y/Bdm and
^/(Jdv, if du and dv are taken positive. The direction parametm of
the tangents to the two curves (m) and (r) are ftc/^ ^/^ H/du
and cx/dv, dy/dv, dzj^v, respectively. Hence the angle a
them is given by the formula
cos a •=■
A^^m
whence sin a — ^EG — F*/VeG. Forming the prodnet mentioned,
we find the same expression as that given above for the element of
area. The formula for cos a shows that F = when and only when
the two families of curves (u) and (r) are orthogonal to each other.
When the surface S reduces to a plane, the formula just found
reduce to the formuUe found in $ 128. For, if we set f{u, r) ■■ 0,
we find
-(I-:)"-©" -gif^UK- "-m'^Ci)'
276
DOUBLE INTEGRALS
[VI, § 132
whence, by the rule for squaring a determinant,
• dx dx
du dv
2
E F
dj d_i
du dv
F G
^^EG- F^
Hence ^EG — F* reduces to |A|w
Exampleg. 1) To find the area of a region of a surface whose equa-
^^ Ig g =/(x, y) which projects on the xy plane into a region R in
which the function f(x, y), together with its derivatives p = df/dx and
q = df/dyy is continuous. Taking x and y as the independent vari-
ables, we find F = 1 -f jo^ F=pq, G=l-{- q% and the area in ques-
tion is given by the double integral
(30) A =/jryrTyT7<^<^. =/X,S^'
where y is the acute angle between the « axis and the normal to the
surface.
2) To calculate the area of the region of a surface of revolution
between two plane sections perpendicular to the axis of revolution.
Let the axis of revolution be taken as the z axis, and let z =f(x)
be the equation of the generating curve in the xz plane. Then the
oodrdinates of a point on the surface are given by the equations
x = pcos<i), 2/ = /osin(i), z=f(p)j
where the independent variables p and w are the polar coordinates of
the projection of the point on the xy plane. In this case we have
ds^ = dp^ll + f>\p)}-{-p^d<o^,
E^l + f'\p), F=0, G = p\
To find the area of the portion of the surface bounded by two plane
•ections perpendicular to the axis of revolution whose radii are pi and
Pf, respectively, p should be allowed to vary from pi to pa (pi< P2) and
M from tero to 27r. Hence the required area is given by the integral
A = jT dpj^ p v/l + /'\p)rf<u = 27rJ'^p^l+f'\p)dp,
tod can therefore be evaluated by a single quadrature. If s denote
the are of the generating curve, we have
VI, J 133] IMPROPER INTEGRAtfi 277
and the preceding formula may be written in the form
J I 2'rrp(U,
The geometrical interpretation of this result it etty: 2wpd* it
the lateral area of a frustum of a cone whoee lUni height U d$ and
whose mean radius is p. Replacing the area between two tiH?tfffPt
whose distance from each other is infiniteaimal by the Utenl aiva
of such a frustum of a cone, we should obtain preciaelj the above
formula for A.
For example, on the paraboloid of revolution generated by revolv-
ing the parabola x^ = 2pz about the z axis the area of the teotum
between the vertex and the circular plane section whoee radius is r is
m. GENERALIZATIONS OF DOUBLE INTEGRALS
IMPROPER INTEGRALS SURFACE INTEGRALS
133. Improper integrals. Let f(x, y) be a function which is
tinuous in the whole region of the plane which lies outside a
contour T. The double integral of /(x, y) extended over the regioo
between F and another closed curve C outside of F has a finite valuo.
If this integral approaches one and the same limit no matter how
C varies, provided merely that the distance from the origin to the
nearest point of C becomes infinite, this limit is defined to be the
value of the double integral extended over the whole regioo
outside F.
Let us assume for the moment that the funotion /(*, y) has a
constant sign, say positive, outside F. In this case the limit of the
double integral is independent of the form of the curves C For,
let Cj, C,, • • •, C., • • • be a sequence of closed curves each of whieb
encloses the preceding in such a way that the distance to the nearesi
point of C\ becomes infinite with n. If the double integral /, extended
over the region between F and C, approaches a limit /, the same will
be true for any other sequence of curves C{, Ci, •••, C^, ••• which
satisfy the same conditions. For, if /I, be the value of the double
integral extended over the region between P and C^, n may be
chosen so large that the curve C, entirely enoloees C^, and w%
shall have l'<K< /. Moreover /I increases with ak Uenee il
278 DOUBLE INTEGRALS [VI, §133
hat a limit /* < /. It follows in the same manner that I < V. Hence
/* =s /, i.e. the two limits are equal.
Ab an example let us consider a function /(x, y), which outside a
eiiiJe of radius r about the origin as center is of the form
where the value of the numerator ^(a;, y) remains between two posi-
tive numbers m and M. Choosing for the curves C the circles
ooDoentrio to the above, the value of the double integral extended
over the circular ring between the two circles of radii r and R is
given by the definite integral
i""X'
^(p cos (u, p sin Q>)p dp
p"
It therefore lies between the values of the two expressions
2-J"^. 2.Mf
dp
p2«-l"
By $ 90, the simple integral involved approaches a limit as R
increases indefinitely, provided that 2a — 1 > 1 or a>l. But it
becomes infinite with R if a<l.
If no closed curve can be found outside which the function f(x, y)
has a constant sign, it can be shown, as in § 89, that the integral
!ff{Xy y)dxdy approaches a limit if the integral Jf\f(x, y)\dxdy
itself approaches a limit. But if the latter integral becomes infinite,
the former integral is indeterminate. The following example, due
to Cayley, is interesting. Let f{x, y) = sin (x^ + y^), and let us inte-
grate this function first over a square of side a formed by the axes
and the two lines x = ayy = a. The value of this integral is
M^'
%\T\{7*'\-y^dy
%mx*dx X I cosy^dy+j C08x^dx x I sin y^dy.
Am a inereases indefinitely, each of the integrals on the right has
a limit, by f 91. This limit can be shown to be V7r/2 in each case ;
hence the limit of the whole right-hand side is tt. On the other
band, the double integral of the same function extended over the
quarter circle bounded by the axes and the circle x^ 4- y^ = R* is
equal to the expression
VI, flWj IMPROPKK IVTFriRUs ff^
which, as R becomes infinite, osciUatat between lero tnd v/2 and
does not approach any limit whatever.
We should define in a similar manner the double infeegral of a
function /(;r, y) which becomes infinite at a point or all along a line.
First, we should remove the point (or the line) from the field of
integration by surrounding it by a small contour (or by a eooUwr
very close to the line) which we should let diminish indefinitely.
For example, if the function J{x, y) can be written in Um fom
/(x, y) = ^^>y)
in the neighborhood of the point (a, b), where f (ar, y) liea twtwseu
two positive numbers m and .U, the double integral of /(», y)
extended over a region about the point (a, h) which contains no
other point of discontinuity has a finite value if and only if a is
less than unity.
184. The function B(p, q). We have .i -;: .. 1 i' . • v C,
recedes indefinitely in every direction. H t r . . :. . , jp.
pose that only a certain portion recedes to infinity. Thiji U tht cait
example of Cayley 's and also in the following example. Lei as take
/(x, y) = 4aE«J'-»y««-»«— '-»•,
where p and q are each positive. This funcUon is continoous sad podtiTe in tke
first quadrant. Integrating first over the square of side • boonded 1^ Um aas
and the lines x = a and y = a, we find, for the value ol the doable laiefial,
r"2x«i»->e-«'dx X r*2y«t-»«-i'4r.
Each of these integrals approaohes a limit as a becomes inflnlle. For. by ihm
definition of the function T{p) in $ 92,
r(p)= fV-Je-'dC,
whence, setting i = x*, we find
(31) r(p)= r**ax«i-»e-'*di.
Hence the double integral approaches the limit r(p)r(9) as a beooflM
Let us now integrate over the quarter circle boonded by tiM ans aad I
circle x^ + i^ = I{*. The value of the double iategral In polar eofltdiaalea Is
2gO DOUBLE INTEGRALS [VI, §135
Ai R becomei Infinite this product approaches the limit
T(p-\-q)B(Py q),
wbtn w« hATe aet
w
^^ B(p, q)=f*2 cosap-i an^q-i d<t> .
liyiiMlm the fact that these two limits must be the same, we find the equation
(88) * T{p) nq) = T{p + q) B(p, q).
The Intagral B(p, q) U called Euler's integral of the first kind. Setting t = sin^ 0,
it may be written in the form
(84) B(p,q)=f\^-Hl-ty-'dt.
The formula (83) reduces the calculation of the function B{p, q) to the calcu-
laUon of the function r. For example, setting p = q = 1/2, we find
whoDoe r(l/2) = Vjt. Hence the formula (31) gives
2
In general, setting q = l-p and taking p between and 1, we find
X^/i-ty-^dt
\~r) J
We ■hall Me later that the value of this integral is TC/smpje.
Itf. tarface Integrals. The definition of surface integrals is analogous to that
of line integrala. Let S be a region of a surface bounded by one or more curves r.
We ahall aesome that the surface has two distinct sides in such a way that if one
side be painted red and the other blue, for instance, it will be impossible to pass
from the red tide to the blue side along a continuous path which lies on the sur-
fiioe and which does not cross one of the bounding curves.* Let us think of S as
a nftteiial torface having a certain thickness, and let m and m' be two points
QMur eaeh other on opposite sides of the surface. At m let us draw that half of
the DomuU mn to the surface which does not pierce the surface. The direction
thof dtlfaMd i4>on the normal will be said, for brevity, to correspond to that side
of Um Mr&oe on which m lies. The direction of the normal which corresponds
to the other tide of the surface at the point m' will be opposite to the direction
joitdtflned.
Lst s s ^(x, y) be the equation of the given surface, and let <S be a region of
this furfioo booaded by a contour r. We shall assume that the surface is met
fft at SMMl one point by any parallel to the z axis, and that the function 0(x, y)
* It li Tety eaay to form a sarfaoe which does not satisfy this condition. We need
miif doform a rectangular sheet of paper ABCD by pastiuK the side BC to the side AD
la tMh A way that the polot C ooincidee with A and the point B with 1).
yi,im] 8UBFACE IKTE6RALS og|
is continuouB inside the region il of the xy pUoe wnvQA 14 bPttfided bj ihM eime C
into which r projecU. It is eTldent that this MTteot hM two ^ ts for mhkh
the corresponding directions of the normal make, iwpMtifely, aesta and
angles with the positive direction of the z axis. We ahall call that aid*
corresponding normal makes an acate angle witli the poaHlva a ■**■ the
aide. Now let P(x, ]/, z) be a function oA the three rariablea s, y, and i whkk
is continuous in a certain region of space which oontaina the refkMi B of Ihaavw
face. If z be replaced in this function by ^(z, y), there reaalu a eertalo foaeCkiB
P [z, y, ^(z, v)]oix and y alone ; and it is natural by analogy with Um intagnito
to call the double integral of this function extended orer the regioa A,
(^&) ff P[x.y,^(*,y)]dxdy.
the surface integral of the function P{z, y, t) taken oTer the regiioo 5 of tha glrca
surface. Suppose the coordinates z, y , and z of a point of 8 giren in term ctf two
auxiliary variables u and v in such a way that the portion 5 of the aufaee con^
sponds point for point in a one-to-one manner to a region A of the «t plane. Lei
d<r be the surface element of the surface 8, and 7 the acute angle baHwoaa Um poii-
tive z axis and the normal to the upper side of S, Then the p TrH ' n g doohla
integral, by §§ 131-132, is equal to the double integral
(36) C C Fix, y, z) cos -r«l<r
where z, y, and z are to be expressed in terms 01 <« auu v. Thia nowi
is, however, more general than the former, for coa7 may take oa either o< two
values according to which side of the surface ia choaea. When the aoiifta aa|^ 7
is chosen, as above, the double integral (35) or (86) ia called the aorfaoe
(37) ffp{x,v,z)dxdy
extended over the upper side of the surface S. liui if > he taken as the obt4iae
angle, every element of the double integral will be changed in aign, and the Dew
double integral would be called the surface integral // Pctsdy ezianded orar tka
lower sideofS. Ingeneral, the surface integral// Pdzdyiaeqtialfo ± tliedosbia
integral (85) according as it is extended over the upper or the lowar aide o( 8.
This definition enables us to complete the analogy between ainple aftd doaUa
integrals. Thus a simple integral changea sign when the Umlu are intetchaafed,
while nothing similar has been developed for double laiegrala. With the gen-
eralized definition of double integrals, we may say that the lntafral///(z, y)d« ^
previously considered is the surface integral extended over the upper aide of the
xy plane, while the same integral with iu sign changed repnaenU the antfaoe
integral taken over the under side. The two senaea of motion for a dai pl e iBl»>
gral thus correspond to the two sides of the xy pUna for a dooblt IntagiaL
The expression (3d) for a surface integral evidently doea noi require that the
surface should be met in at moat one point by any parallal to tiM f axiiL In the
same manner we might define the surface integrate
//g(x, y. «)dycfz, ff^^i^ y» «)dtda,
282
DOUBLE INTEGRALS
[VI, § i3e
and tht more gener&l integral
ffP(x, y, z)dxdy-hQ{x, y, z)dydz + R(x, y, z)dzdx.
Mm hitngnl may also be written in tlie form
r r [P C08 7 + Q cos a + -B COB /3] d<r ,
a, ^, Y aie the direction angles of the direction of the normal which cor
to the side of the surface selected.
Sorfaoe integrals are especially important in Mathematical Physics.
Its. StokM' theorem. Let £ be a skew curve along which the functions
P(x, y, «), Q{x, y, z), iJ(x, y, z) are continuous. Then the definition of the line
integral
Pdx-\-Qdy-\-Rdz
L
taken along the line L is similar to that given in § 93 for a line integral taken
along a plane curve, and we shall not go into the matter in detail. If the curve L
Is closed, the integral evidently may be broken up into the sum of three line inte-
grals taken over closed plane curves. Applying Green's theorem to each of these,
it is evident that we may replace the line integral by the sum of three double
integrals. The introduction of surface integrals enables us to state this result in
Terj compact form.
Let OB consider a two-sided piece S of a surface which we shall suppose for
deftniteneas to be bounded by a single curve r. To each side of the surface
oorreiponds a definite sense of direct motion along the contour F. We shall
BiMime the following convention : At any point M of the contour let us draw
that half of the normal Mn which corresponds to the side of the surface under
concideration, and let us imagine an observer with his head at n and his feet at M)
we shall say that that is the positive sense
of motion which the observer must take in
order to have the region S at his left hand.
Thus to the two sides of the surface corre-
spond two opposite senses of motion along
the contour r.
Let us first consider a region S of a sur-
face which is met in at most one point by
any parallel to the z axis, and let us suppose
the trihedron Oxyz placed as in Fig. 30,
where the plane of the paper is the yz plane
and the x axis extends toward the observer.
To the boundary r of S will correspond a
closed contour C in the xy plane ; and these
two curves are described simultaneously in
the sense indicated by the arrows. Let
the eqaaUon of the given surface, and let P(x, y, z) be a function
which ii conUouout In a region of space which conUins 8. Then the line inte-
Fio.ao
« -P«, y) be
which li oontii
i^* ^n'*^'' V,t)dx\» Identical with the line integral
VI, §ia«] SURFACE INTEGRALS
uken along the plane curve C. Lei at applj Ora«B*a thMtwi (| 1S8) to
latter integral. Setting
?(5ry) = P[«, y, ^x,|f)]
for definiteneas, we find
dy dy Hby"^ 9f 0017*
where a, /3, 7 are the direction angles of the normal to the l^ptr iMt of 0.
Heuce, by Greeirs theorem,
where the double integral ia to be taken over the ngioo A of the «y |4aao
bounded by the contour C. But the right-hand aide ia simply the surtaeo
integral
//(f-«-g»')
extended over the upper side of 8 ; and henoe we may write
f P(x,y,z)dx= r f t^dzdz^^dxdv.
This formula evidently holds also when the surface integral is taken
other side of S, if the line integral Is taken in the other directioti aloof P. Aad
it also holds, a^ does Green> theorem, no matter what the foiw of tke
may be. By cyclic permutation of x, y, and z we obtain the fol
formulee :
Adding the three, we obuin Stokm'^ lAaorcm to iU gm§nifonm :
f P{x, y, x)dz + Q(x, y. t)dy + «(«, y, i)df
(88)
The sense in which r is described and the side of tbe eafteoe over wlOeii
double integral is taken correspond aooording to tbe eoateslkHi
2g4 DOUBLE INTEGRALS rvi,§i37
IV. ANALYTICAL AND GEOMETRICAL APPLICATIONS
137. yolmnM. Let us consider, as above, a region of space bounded
by the xy plane, a surface 5 above that plane, and a cylinder whose
generators are parallel to the z axis. We shall suppose that the
section of the cylinder by the plane « = is a contour similar to
that drawn in Fig. 25, composed of two parallels to the y axis and two
oorvilinear arcs A PB and A 'QB'. li z = f{x, y) is the equation of the
surface S, the volume in question is given, by § 124, by the integral
A^y y)dy.
Now the integral C*f(^y y)^y represents the area A of a section of
this volume by a plane parallel to the yz plane. Hence the preceding
formula may be written in the form
(39) F=jr
b
Adx,
The volume of a solid bounded in any way whatever is equal
to the algebraic sum of several volumes bounded as above. For
instance, to find the volume of a solid bounded by a convex closed
surface we should circumscribe the solid by a cylinder whose gen-
erators are parallel to "the z axis and then find the difference between
two volumes like the preceding. Hence the formula (39) holds for
any volume which lies between two parallel planes x = a and x = b
(a < b) and which is bounded by any surface whatever, where A
denotes the area of a section made by a plane parallel to the tv/o
given planes. Let us suppose the interval (a, b) subdivided by the
points a, x^ a;,, • • -, a;^_,, by and let Ao, Aj, • • •, Af, • • • be the areas
of the sections made by the planes x = a, x = Xi, • • • , respectively.
Then the definite integral j\ dx is the limit of the sum
Ao(Xi - a) -f- Ai(x5, _ xi) + . . . + Ai_i(x,. - a;,._i) . • -.
The geometrical meaningof this result is apparent. For A,_i (a;,. — a;,_i),
for instance, represents the volume of a right cylinder whose base is
the section of the given solid by the plane x = ar,_i and whose height
is the distance between two consecutive sections. Hence the volume
of the given solid is the limit of the sum of such infinitesimal cylin-
ders. This fact is in conformity with the ordinary crude notion of
volume.
VI, $ 138] APPLICATinVS
S86
If the value of the area A be known as a function of x, the toI-
ume to be evaluated may be found by a single quadrature. As aa
example let us try to find the volume of a portion of a •olid of evo-
lution between two planes perpendicular to the axis of rerolutloiL
Let this axis be the x axis and let « «/(x) be the equalkm of tkm
generating curve in the xz plane. The seolioo made by a pUuie par-
allel to the yz plane is a circle of radius f{x), Henoe the leqoired
volume is given by the integral 'rrj^lf{x)Ydx.
Again, let us try to find the volume of the portion of the ellipsoid
ar* V* «•
—-4- ^-4- —= 1
a« ^ 6« ^ c« ^
bounded by the two planes a; = a^o, « = X. The section made by a
pla ne parall el to th e plane x = is an ellipse whose semiaxes are
b Vl-xya=» and c Vl - x^/a\ Hence the volume sought is
To find the total volume we should set atp= — a and JT as a, which
gives the value ^Trabc.
138. Ruled surface. Phsmoidal formuU. When the 4rea A is aa iatsgnl
function of the second degree in x, the volume nuy be expiwnd Tery dapiy
in terms of the areas B and W of the bounding aectioiis, the ana 6 of the mmm
section, and the distance h between the two boondiag ^«»f fAn f If the bmb
section be the plane of yz^ we have
r= r^"(te* + 2mx + n)dx = 2Z^ + »iio.
J-a 8
But we also have
A = 2a, 6 = n, B = Ufi -i- 2ma -^ n^ IT = (a* - tRUi -f n,
whence n = h,a = A/2, 2ta* = B -f B" - 26. Theee equations kad to tke formaiA
(40) r=:^[fi+B' + 461.
which is called the prismoidai formula.
This formula holds in particular for any soUd bduidtd by a
two parallel planes, including m a qwcial caae the so-calkd prismokL*
\ety = ax -\- p and z = to + 9 be the equations of a variable smiglil Bne,
a, b, p, and q are continuous functions of a variable paruneter f whieb
their initial values when t incre— ei frcHn l« to T. Tids stntsht Uae
* A prismoid \h a solid bounded by aoy
lei aud coutaiu all the vertices. — TaAits.
286
DOUBLE INTEGRALS
[VI, § 139
a ruled surftwe, and the area of the section made by a plane parallel to the plane
s = is giren, by S 94, by the integral
A= r {ax-^p){b'x-hq')dt,
wber« a', fc', C, d' denote the derivatives of a, 6, c, d with respect to t. These
derivaUves may even be discontinuous for a finite number of values between to
and r, which will be the case when the lateral boundary consists of portions of
•everal ruled surfaces. The expression for A may be written in the form
A = a;« f ab'dt + x f {aq' -\- pb')dt + f pq'dt,
where the integrals on the right are evidently independent of x. Hence the
formula (40) holds for the volume of the given solid. It is worthy of notice that
the Mame/ormiUa also gives the volumes of most of the solids of elementary geometry.
189. Viviani's problem. Let C be a circle described with a radius OA {= R)
of a given sphere as diameter, and let us try to find the volume of the portion
of the sphere inside a circular cylinder whose right section is the circle C.
Taking the origin at the center of the sphere, one fourth the required volume
is given by the double integral
- = / / Vi?2 - a;2 - y2 dxdy
extended over a semicircle described on OA as diameter. Passing to polar coor-
dinates p and w, the angle u varies from to 7t/2, and p from to E cos w. Hence
we find
Fto.81
— = - I (K» - i28 sm« w) dw = — I I •
4 8 Jo ^ ' 3 V2 3/
If this volume and the volume inside the cylinder
which is symmetrical to this one with respect to
the z axis be subtracted from the volume of the
whole sphere, the remainder is
3
nR^
8i?» /tt 2\ _
3 V2 3
!\_ 16
R«
Again, the area of the portion of the sur-
face of the sphere inside the given cylinder is
= 4 jy VrTp«Tg* dx dy .
pwxiAqhf their valuM - z/t and - y/«, respectively, and passing to
polar eottrdtnaUa, we find
VI, f 140] APPLICATIONS f87
or
= 4K»J''(1 - tin •-)di# = 4IP /- - iV
Subtracting the area encloeed by the two ojlinden fiom Um wboto an* of tko
sphere, the remainder ia
140. Evaluation of particular definite integrala. The tbeorOBf ettal^
lished above, in particular the theorem regarding diifereoiuiiUNi
under the integral sign, sometimes enable us to evaluate oeitain defi-
nite integrals without knowing the corresponding indefinite integrals
We proceed to give a few examples.
Setting
the formula for differentiation under the integral sign giret
Id ^ log(t-f g*) f xdx
da" 1-ha* J. (l + «5)(l-fO*
Breaking up this integrand into partial fractions, we find
X 1 / J 4- *r _ g \
whence
r xdx _ __ log(l4-a*) ■ tf ,nitani»
X (l+ax)(l + x«) 2(1-Ha«) "^l+o*
It follows that
dA a ^ . log(l-f<t*)
-r— =:r ; aTC tan a 4- 0.4 . -^ '
whence, observing that A vanishes when a as 0, we may write
Integrating the first of these integrals hj parts, we finally find
A =rarctanalog(l-h«*)-
2gg DOUBLE INTEGRALS [VI, §140
Again, consider ihe function x". This function is continuous
when x'lies between and 1 and y between any two positive
numbers a and h. Hence, by the general formula of § 123,
J\ dx\ x^dy = j dy I x^dx.
But ^ ,
hence the value of the right-hand side of the previous equation is
On the other hand, we have ^
whence &,.UU^m. ^
^..^
i^T^"^^=^^^(^i)- ^ '(^^
/r ^.r.'^.Mj^i^
In general, suppose that P(a5, y) and Q(x, 2^) are two functions \j^ -^
which satisfy the relation dPjdy = dQ/dx, and that iCo, a^i, 2/o> 2/1 are / u
given constants. Then, by the general formula for integration
under the integral sign, we shall have
or
[P(x, y,) - P(x, yo)]<«» = I [Q(a^i, y) - Q{x,, y)] c^y.
Cauchy deduced the values of a large number of definite inte-
grals from this formula. It is also closely and simply related to
Green's theorem, of which it is essentially only a special case.
For it may be derived by applying Green's theorem to the line
integral jPdx -\- Qdy taken along the boundary of the rectangle
formed l^ the lines jt = iCo, « = acj, y = yo, y = yi-
In the following example the definite integral is evaluated by a
•pecial derioe. The integral
F(a) = / log (1 — 2a008x-^a^dx
VI. i 140] APPLICATIONS f§§
has a finite value if \a\ is different from uni^v This fonelioo
F(a) has the following properties.
1) F(- a) = F{a). For
Fi^-a)^! :og(l4-2aooi« + «•)<{*,
or, making the substitution x = ir — y^
^(-«)=/ log(l-2acosy + a«)rfy«F(a).
2) F(a«) = 2F(a). For we may set
2F(a) = /^a) + F(-a),
whence
2F(ar) = J [log(l - 2a cos X 4- a*) + log (1 4- 2a 008 z + a*)]***
= 1 log(l-2a^cos2z + a*)«ic.
If we now make the substitution 2x = y, this becomas
•2F(a) = iJ^ log(l-2a«c08y-f a*)rfy
-f ^J^ log(l- 2a« cosy 4- a*)<fy.
Making a second substitution y = 2tr — r in the last integral, w*
find
r log(l- 2^« cosy 4- a*)rfy=r log(l- 2a«oos« 4- a*)rf«,
which leads to the formula
From this result we have, successively,
F(«) = |F(a«) = Jfr«*)=.. = iF(aO.
If I a I is less than unity, <r** approaohes wsto as fi beoomet m^ouMi
The same is true of F{it^), for the logwithm approtolMi
Hence, if I or I < 1, we have F{a) = 0.
iM DOUBLE INTEGRALS [VI, §141
If |a| U greater than unity, let us set a = 1/^. Then we find
''•)-x;-('-T^4)-
= r log (1-2/3 cos x-f-)8')c?a;-7r log )8«,
where |/5| is less than unity. Hence we have in this case
F{a) = - TT log^ = TT log a^.
Finally, it can be shown by the aid of Ex. 6, p. 205, that F{± 1) = ;
hence F(a) is continuous for all values of a.
141. Approxinute value of logr(n + 1). A great variety of devices may be
employed to find either the exact or at least an approximate value of a definite
Integral. We proceed to give an example. We have, by definition,
r(n + l) = r x'^er^dz.
Jo
The function x*c-* assiimes its maximum value n^e-" f or x = n. As x increases
from zero to n, af*c-* increases from zero to n^c-" (n>0), and when x increases
from n to + «, x*e-* decreases from n^e-" to zero. Likewise, the function
n^er*e-^ increases from zero to n^e-" as t increases from — oo to zero, and
decreaoes from n"e-'« to zero as t increases from zero to + oo. Hence, by the
rabetitation
(42) x"e-* = n"e-«e-«",
the values of x and t correspond in such a way that as t increases from — oo
to + 00, z increases from zero to + oo.
It remains to calculate dx/dt. Taking the logarithmic derivative of each side
of (42), we find
dx_ 2tx
dt~ x — n
We have also, by (42), the equation
<a = X -n -nlog (- j
For simplicity let ua §et x = n + 2, and then develop log(l + z/n) by Taylor's
theorem with a remainder after two terms. Substituting this expansion in the
nXvm for (*, we find
L~ 2n«(l+<?|fJ 2{n + dz)^
9 liaa between zero and unity. From this we find, successively,
r^.=K"^0 -[%!-<>-"]•
VI, 5 142J APPUCAT10N8 291
whenoe, applying the fommJa for obaoga of Tariabl*,
r(n + 1) = 2n-<- ^ f^^^^"^ + 2»-e- J** V^<U !)!<«.
The flret integral U
At for the lecond integral, though we cannot eTaloate it exactly, iIdm «• do
not know 0, we can at least locate iu value between oeruin And t f^H i f^
all its elementA are negative between — oo and nro, and they an all pa^^ihm
between zero and + oo. Moreover each of the integrmla /* , /^* la lea Ib
absolute value than //* te-*'flM = 1/2. It follows that
(43) Tin + 1) = v^ tfr^/vi + -^V
where la lies between - 1 and + I.
If n is very large, u/y/2n is very small. Hence, if we take
r(n + 1) = n-e-^V^ni
as an approximate value of r(n + !)« our error is reUtively small, tboogb tka
actual error may be considerable. Taking the logarithm of each aide of (4S), w
find the formula
(44) log r(n + 1) = (n + 1) logn - n + 1 log(Sr) + 1.
where e is very small when n is very large. N^ecting «, we hare an
which is called the asymptx>tic value of logr(n + 1). TTiis formala la
esting as giving us an idea of the order of magnitude of a f^oCorfal.
142. D'AIembert'8 theorem. The formula for integration under the
sign applies to any function /(z, y) which is continnous in the rsetangit of fail*-
gration. Hence, if two different results are obtained by two diflTerenl w alhods
of integrating the function /(x, y), we may conclude that the foneCkai /(a, y) la
discontinuous for at least one point in the field of integration. GaiMB dadncad
from this fact an elegant demonstration of d*Alembert*s tbaoram.
Let F{z) be an integral polynomial of degree m in s. W« shall aanaa for
definiteness that all its coefficients are real. Replacing s by ^ooat* -f idaw),
and separating the real and the imaginary parts, we have
F(«) = P + <Q.
where
P = Aop'^ccmnua -{■ AifF^'^CMim - !)•# + ••• +il«,
Q = i4o^sinm« + Ji^-» sln(m - 1)m + • •• -i-ila.t^alnw.
If we set 1^ = arc tan (P/Q), we shall have
ap~P«+Q«' d*» F«+^*
and it is evident, without actually carrying out th« calculation, thai the
derivative is of the form
bpd^ (P« + W
292 DOUBLE INTEGRALS [VI, Exs.
^,„^ jf It ft conUnuoua function of p and «. This second derivative can only
be dtoconlinuouB for values of p and w for which P and Q vanish simultaneously,
that is to My, for the rooU of the equation F{z) = 0. Hence, if we can show that
the two iDteg^rals
an nnaqoal for a given value of iJ, we may conclude that the equation F{z) =
has at least one root whose absolute value is less than B. But the second inte-
gral is always zero, for
Jf ^"i^Z; da, = f— T " ^^ »
and bV/bp is a periodic function of w, of period 2;f. Calculating the first inte-
gral in a similar manner, we find
Jq 5p3w LS"Jp=o
and it is easy to show that dV/d(a is of the form
du> ~ AIp^^-\-"'
le degree of the terms not written down is less than 2m in p, and where
tha numerator contains no term which does not involve p. As p increases indefi-
nitely, Uie right-hand side approaches — m. Hence R may be chosen so large
that the value of dV/du, for p = iJ, is equal to — m + e, where c is less than m
tat abeolute value. The integral /(,^"'(- m + c)dw is evidently negative, and
the first of the integrals (45) cannot be zero.
EXERCISES
1. At any point of the catenary defined in rectangular coordinates by the
equation
= |(ei + .-i)
let OS draw the tangent and extend it until it meets the z axis at! a point T.
Reroliing the whole figure about the x axis, find the difference between the areas
deeeribed by the arc ilAf of the catenary, where A is the vertex of the catenary,
and that deeeribed by the tangent MT {\) as a function of the abscissa of the
poinl M, (8) as a function of the abscissa of the point T.
[Licence^ Paris, 1889.]
f. Using the usual system of trirectangular co5rdinates, let a ruled surface
to formed as follows : The plane zOA revolves about the x axis, while the gen-
WAtiBf line X>, which lies in this plane, makes with the z axis a constant angle
whose taogent is X and ouU off on OA an intercept OC equal to \ad, where a
If a fiteo leofth and # is the angle between the two planes zOx and zOA.
VI, En.] EXERCISES
1) Find the Tolame of Um aoUd bounoed by iha nil«d fUifAot aod th»
xOy, zOx, and zOA, where the angle $ between ihe laei two is Itm * ^n j
2) Find the area of the portion of the aorfaoe boonded bj the planet sOy
xOx, zOA.
[Liemf, Parit, July, 1881.]
3. Find the volumo of the solid boonded by the zy plana, iha eylladar
6^x^ + a'^y^ = a^6^, and the elliptic paraboloid wboae eqoatioa in rnHiiiiiilai
coordinates is
[Lkmee, Paiia, lltt.]
4. Find the area of the curvilinear quadrilateral bounded by the (bar
focal conicfl of the family
which are determined by giving X the values c«/8, S^/S, 4^/S, 6eV8, i
[Lketiet, H—inyn, 18B6.]
5. Consider the curve
y = V2(8inx ~oosz),
where x and y are the rectangular co<)rdinatet of a point, and
from it/i to 67r/4. Find :
1) the area between this curve and the z axis ;
2) the volume of the solid generated by revolving the eunre about the x axis ;
3) the lateral area of the same solid.
[Lie€me€, Montpellter, 18ea.]
6. In an ordinary rectangular coordmaie plane let A and B be any two
points on the y axis, and let .1 Mli be any curve Joining A and B whteh, tQfMter
with the line AB, forms the boundary of a region AMBA wliOM araa la a pi<a-
assigned quantity S. Find the value of the following daflnite latagnl
over the curve A MB :
f[<f>{v)e' -my]dx-{- [^'(jf)«« - Hdr .
where m is a constant, and where the function ^(y), together with tti dMivttlvi
<P'iy), is continuous.
[Xieenet, Naney, IMi.]
7. By calculating the double integral
/ e-*v8inaxdifd:z
in two different ways, show that, provided that a is not lero,
"••^«d. = ±l
r^*sin«
8. Find the area of the lateral surface of the portion of an sUlpsoid of revo-
luUon or of an hyperboloid of revolntioo whkh Is boondsd bf two planes
dicular to the axis of revolution.
S94 DOUBLE INTEGRALS [VI, Exs.
9*. To ittd the area of an ellipsoid with three aneqaal axes. Half of the total
ax«ft A ia given by the double integral
'■■ffV^^W
dxdy
orer the interior of the ellipse b»x^ + a^y^ = a'^V^. Among the methods
employed to reduce this double integral to elliptic integrals, one of the simplest,
due to Catalan, consists in the transformation used in § 125. Denoting the
integrand of the double integral by v, and letting v vary from 1 to + oo, it is
eftqr to show that the double integral is equal to the limit, as I becomes infinite,
ol the difference
najb
This expression is an undetermined form ; but we may write
/
v^dv
V(— S)(— 8
c2\ L
V
f-tM
and hence the limit considered above is readily seen to be
irab
■{'-i)i-S)f
dv
W». If from the center of an ellipsoid whose semiaxes are a, 6, c a perpen-
* T be let fall upon the tangent plane to the ellipsoid, the area of the surface
la the locoa of the foot of the perpendicular is equal to the area of an
•IHpiokl whose aeniiaxea are bc/a, ac/b, ab/c.
[William Kobkkts, Journal de Liouville, Vol. XI, 1st series, p. 81.]
VI, Ex. ] KXERCI8E8
11. Evaluate the double integral ut the ezpre
extended over the interior of the triangle boonded bjr tbt itralflit Iteea y « a«,
y = X, and z = X in two different ways, and tberebj wtabliah tb« formula
/'dx/"(»-y)-/(y)dy= f ^^'^^^' m^-
lilt deduce the relation
f dzf dx " f Ax)dx = r (X - yrJ\y)uy.
From this result deduce the relation
In a similar manner derive the formula
C'xdz f'xdx- f'xdx f'Ax)dz = __1_^ r V - l^)-/Ur)^.
and verify these formulas by means of the law for diflamltatloB nadir tiM
integral sign.
CHAPTER VII
MULTIPLE INTEGRALS
INTEGRATION OF TOTAL DIFFERENTDLLS
I. MULTIPLE INTEGRALS CHANGE OF VARIABLES
143. Triple integrals. Let F{Xj y, «) be a function of the three
Tariables x, y, z which is continuous for all points M, whose rec-
tangular coordinates are {x, y, z), in a finite region of space {E)
bounded by one or more closed surfaces. Let this region be sub-
divided into a number of subregions (^i), (eg), ••, (««), whose vol-
umes are w„ V,, •••, v^, and let (^,-, 17,, Q be the coordinates of any
point m^ of the subregion (e,). Then the sum
(1) XK^i, -rn^^d^i
1=1
approaches a limit as the number of the subregions (e,) is increased
indefinitely in such a way that the maximum diameter of each of
them approaches zero. This limit is called the triple integral of
the function F{xy y, z) extended throughout the region (E)^ and
is represented by the symbol
(2) / / / F{x,y,z)dxdydz.
J J J(,E)
The proof that this limit exists is practically a repetition of the
proof giren above in the case of double integrals.
Triple integrals arise in vai'ious problems of Mechanics, for
instance in finding the mass or the center of gravity of a solid
body. Suppose the region (E) filled with a heterogeneous sub-
ftanoe, and let fi(x, y, z) be the density at any point, that is to say,
the limit of the ratio of the mass inside an infinitesimal sphere about
the point (x, y, «) as center to the volume of the sphere. If /oti and /aj
are the maximum and the minimum value of /x in the subregion (e,),
it \M evident that the mass inside that subregion lies between /ajV,
and mv<; henoe it is equal to v./i(^„ ,;,, Q, where (^,, 7;., Q is a
suitably choeen point of the subregion («,). It follows that the total
296
VII. $143] INTUODUCTION CHANGE OF VABIABLBS 297
mass is equal to thf^ triple integral ffffidxdpdm eitended Ihroofh-
out the region (E).
The evaluation of a triple integral maj be reduced to Ibe fo^
cessive evaluation of three simple integraU. Let ua auppoee first
that the region (E) is a rectangular parallelepiped bounded by the
six planes a; = jto, a; = ,Y, y = 5/0, y «= K, «■»«»,« — Z. h^ (K)
he divided into smaller parallelopipeda by planet parallel to the
three coordinate planer. The volume of one of the latter ia
(^i — ar,-,) (y^ — y^_,) (zi — «|_i), and we have to find the limit ol
the sum
i L I
where the point ((,iif Vutf Cm) i^ ^".v jxhmi 11 .r.' ,
parallelopiped. Let us evaluate tirst tliat ]- ^ .si... i. .i-. .-.
from the column of elements bounded by the four plaoee
x = Xf_,, r = Xi, y = y»-i, y = y»,
taking all the points (^^.,, rim, Cm) upon the straight line x bx^.,,
y = yt-i' This column of parallelopipeds gives rise to the turn
(x, - Xi_,)(y, - yt-,)[F(x,_,, y»_,, Ci)(«i -«•) + •••]»
and, as in § 123, the ^'s may be chosen in such a way that the
quantity inside the bracket will be equal to the simple integtal
It only remains to find the limit of tiie sum
• k
But this limit is precisely the double integral
<P{Xy y)dxdy
extended over the rectangle formed by the linee « » «#, 9 « X,
y — y(i>y= Y. Hence the triple integral is equal to
C dxf ♦(X, y)dy,
or, replafincr ^^/'.r. »/) by its value.
(4) / ^^i "^J ''^'» y» *''*^-
//•
208
MULTIPLE INTEGRALS
[VII, § 144
The meaning of this symbol is perfectly obvious. During the first
integration x and y are to be regarded as constants. The result will be
a function of x and y, which is then to be integrated between the limits
y« and K, X being regarded as a constant and y as a variable. The
result of this second integration is a function of x alone, and the last
step is the integration of this function between the limits Xq and X.
There are evidently as many ways of performing this evaluation
as there are permutations on three letters, that is, six. For instance,
the triple integral is equivalent to
' dz\ dx f F(x, y, z)dy = j ^(z)dz,
where ♦(«) denotes the double integral of F(x, y, z) extended over
the rectangle formed by the lines x = Xq^ x = X, y = y^, y = Y. We
might rediscover this formula by commencing with the part of the
sum S which arises from the layer of parallelopipeds bounded by the
two planes z = «^_i, z = Zi. Choosing the points (^, -q, ^) suitably,
the part of S which arises from this layer is
and the rest of the reasoning is similar to that above.
144. Let us now consider a region of space bounded in any
manner whatever, and let us divide it into subregions such that any
line parallel to a suitably chosen
fixed line meets the surface which
bounds any subregion in at most
two points. We may evidently
restrict ourselves without loss of
generality to the case in which a
line parallel to the z axis meets
the surface in at most two points.
The points upon the bounding
surface project upon the xy plane
into the points of a region A
bounded by a closed contour C.
To every point {x, y) inside C cor-
respond two points on the bound-
ing surface whose coordinates are
We shall suppose that the functions
Let us now
Fio. 32
•i - ♦iC*! y) and «, s ^(x, y).
^1 and ^ are continuous inside C, and that <^i < </>
VII, §144] INTRODUCTION CHANGE OP VARlAIUFs f^q
divide the region under consideration i)y pUmcs panUiei to the ooflf-
dinate planes. Some of the subdivisioni will be portiMii of ptnl-
lelopipeds. The part of the sum (1) which ariiat from the ooIiudd
of elements bounded by the four planee z « x^^i, « a x^, y ■■ y^.,,
y = y^ is equal, by § 124, to the ezprestioD
(^i - «<-i)(y* - y*-i) l^jT F(af,.„ y».„ M)dM -h ^\
where the absolute value of c^ may be made less than any promifncd
number c by choosing the parallel planes sufficiently near together.
The sum
• k
approaches zero as a limit, and the triple integral in question ij
therefore equal to the double integral
J JiA)
*(x, y)dxdy
I
extended over the region {A ) bounded by the contour C, where the
function 0(2;, y) is defined by the equation
*(^, y) =J n^y y, ')dx.
If a line parallel to the y axis meets the contour C in at meet
points whose coordinates are y^= ^1(2) and y^^z ^t(x), respectiTelj,
while X varies from x^ to 2:,, the triple integral may also be writteo
in the form
(6)
f dx dy F{x,y,z)d*.
The limits z^ and z^ depend upon both x and y, the limiti yi and |%
are functions of x alone, and finally the limits X| and x, are cons t ants.
We may invert the order of the integrations as for double inte-
grals, but the limits are in general totally different for different
orders of integration.
Note. If ♦(x) be the function of x given by the double integral
n') =f '^yj '^'C' y* *)*'«
800 MULTIPLE INTEGRALS [VII, § 145
extended over the section of the given region by a plane parallel to
the yM plane whose abscissa is a:, the formula (5) may be written
J I *^(x)dx.
This is the result we should have obtained by starting with the
layer of subregions bounded by the two planes x = Xi_Yj a; = x,.
Choosing the points (^, 17, I) suitably, this layer contributes to the
total sum the quantity
♦(«<-i)(aJ<-a;,_,).
KxampU. Let as evaluate the triple integral fffz dx dy dz extended through-
out that eighth of the sphere x^ + y'* + z^ = R^ which lies in the first octant. If
we integrate first with regard to z, then with regard to y, and finally with regard
to z, the limits are as follows : x and y being given, z may vary from zero to
Vfi* — z* - y* ; X being given, y may vary from zero to VR^ — x^ ; and x itself
may vary from zero to R. Hence the integral in question has the value
J J J zdxdydz =J dx J dy J zdz,
we find successively
§X (ija - x« - y^)dy = [I (12^ - x^)y - ^ys]^ = §(^ " ^')^
and It merely remains to calculate the definite integral \S^{B^ - x^^dx, which,
by the subetitution z = i2 cos 0, takes the form
TT
\ f ^R* sin* <f,d<f>.
o Jo
Hence the value of the given triple integral is, by § 116, 7tR*/lQ.
145. Change of variables. Let
be formulffl of transformation which establish a one-to^ne corre-
•pondenoe between the points of the region (E) and those of another
region (^,). We shall think of m, v, and w as the rectangular coor-
dioatet of a point with respect to another system of rectangular
VII, §145] TNTRODUCTIOK nj wav np vaRUBLK*- ^01
coordinates, m ^'eiH»ral rlifTtrrut from the lir*t. If F(x, y, <) U a
continuous function through' mu the ref^ion (E)f we shall alwavi hava
(7)
fff^F(x,y,z)dxdydx
where the two integrals are extended throughout the reguma (K)
and (Ei)j respectively. This is the formula for change of YariaUet
in triple integrals.
In order to show that the formula (7) always holds, we shall
commence by remarking that if it holds for two or more particular
transformations, it will hold also for the transformation obtained by
carrying out these transformations in suoeession, bj the well-knowD
properties of the functional determinant (S 29). If it is applicable
to several regions of space, it is also applicable to the region obtained
by combining them. We shall now proceed to show, as we did for
double integrals, that the formula holds for a transformation which
leaves all but one of the independent variables unchanged, — for
example, for a transformation of the form
(8)
x = x', y = y', « = f(x', y*, «")■
We shall suppose that the two points ^f{xy y, z) and St(x\ y*, r^ an
referred to the same system of rectangular axes, and that a paimllal
to the z axis meets the surface which
bounds the region {E) in at most two
points. The formulae (8) establish a corre-
spondence between this surface and another
surface which bounds the region (K'). The
cylinder circumscribed about the two sur-
faces with its generators parallel to the
z axis cuts the plane « = along a closed
curve C. Every point w of the region A
inside the contour C is the projection of
two points my and rw, of the first surface,
whose coordinates are «» and «,, respectively, and also of two
points m[ and vi[ of the second surface, whose coordinates are «}
and «i, respectively. Let us choose the notation in such a way
that «i < «„ and «{ < «;. The formula (8) transform the point m^
into the point m|, or else into the point m^. To distinguish the
two cases, we need merely consider the sign of df/dz'. If ^/^s' it
©0
rio.ss
802
MULTIPLE INTEGRALS
[VII, § 146
positive, « increases with z\ and the points mx and m^ go into the
points m{ and mi, respectively. On the other hand, if dy^/Jdz' is
negative, x decreases as «' increases, and m^ and m, go into m[ and
ml, respectively. In the previous case we shall have
r F(x, y, z)dz= f F[x, y, ^(a;, y, «')] ^ «?«',
whereas in the second case
F(x, y, «)rf« = -J Fix, y, i/r(a;, y, ^O] ^, ^«'-
In either case we may write
(9) J\x,y,z)dz= £*Flx,y,^(x,y,z')2\^,dz'.
If we now consider the double integrals of the two sides of this
equation over the region A, the double integral of the left-hand side,
11 dxdy \ F(x, y, z)dz,
J Ju) J*.
is precisely the triple integral/// F(ic, y, z) dx dydz extended through-
out the region {E). Likewise, the double integral of the right-hand
side of (9) is equal to the triple integral of
n^\ y\ K^\ y\ ^')]
extended throughout the region (£:'), which readily follows when
X and y are replaced by a;' and y', respectively. Hence we have in
this particular case
J J J{K)
F(aj, y, z) dx dy dz
J J J(E')
^[^',y',^(a;',y',«')]
dz'
dx'dy'dz'.
But in this case the determinant D(x, y, z)/D(x', y', «') reduces to
9if/dM\ Hence the formula (7) holds for the transformation (8).
Again, the general formula (7) holds for a transformation of the
type
(10) m »/(x', y', •') , y = ^(x', y', z'), z^ k\
VU,514fl] INTRODUCTION CHANGE OF VARIABLES
where the variable z remaina unchaoged. We aball tuppoto ^ Ki l
the formulae (10) establish a one-toone oorreapondenoe betwtM
the points of two regions (E) and (K'), and in particular thai the
sections R and R' made in (E) and (^*), respectivelj, bj aaj
plane parallel to tlie xy plane correspond in a one-to-one manner.
Then by the formulae for transformation of double tntegrmls W9
shall have
(11)
/ / ^(«» y,z)dxdy
The two members of this equation are functions of the rariable
z = z' alone. Integrating both sides again l^tween the limits S|
and «a, between which « can vary in the region (E), we find Um
formula
fffF{x,y,z)dxdydM
J J JiE)
(12)
But in this case D{t, y, «)//>(x', y', z) = i\x, y)/D(x', y^. Heooa
the formula (7) holds for the transformation (10) also.
We shall now show that any change of variables whafeem
(13) a;=/(x„yi, «i), y = <^(a:i, yi, «,), s « ^x,, y,, «,)
may be o'btained by a combination of the preceding transformations.
For, let us set x' = x^, y' = y^ z' s z. Then the last equation of
(13) may be written z' = ^(x'f y\ «,), whence «, = ir(x*, y*, «^
Hence the equations (13) may be replaced bv the six e<]uatlons
(14) X =/[x', y\ 7r(x', y\ *')], y = ^[x', y', -
(16) x' = aH, y' = yi, «' = ^^,-^1, yx, «i
The general formula (7) holds, as we have seen, for each of the
transformations (14) and (15). Henoe it holds for the transforma-
tion (13) also.
We might have replaced the general transformatinn (13>. .^^ th**
reader can easily show, by a seciufan-** nf thnv trannform*tion!i *^{
the type (8).
804 MULTIPLE INTEGRALS [VII, §146
146. Element of volume. Setting F(aj, y, «) =1 in the formula (7),
we find
J J JiE) J J JiE^\
Djx, y, z)
D(u, V, w)
dudvdw.
The left-hand side of this equation is the volume of the region (.E:).
Applying the law of the mean to the integral on the right, we find
the relation
(16) V = V,
DLfy <^> «A)
/)(«, V, w)
(f,»»,o'
where Kj is the volume of {EC), and ^, -q, ^ are the coordinates of some
point in {Ex). This formula is exactly analogous to formula (17),
Chapter VI. It shows that the functional determinant is the limit
of the ratio of two corresponding infinitesimal volumes.
If one of the variables m, v, w in (6) be assigned a constant value,
while the others are allowed to vary, we obtain three families of
surfaces, u = const., v = const., w = const., by means of which the
region (E) may be divided into subregions analogous to the paral-
lelopipeds used above, each of which is bounded by six curved faces.
The volume of one of these subregions bounded by the surfaces
(tt), (u 4- du), (r), (v -f dv), (w), (w -f dw) is, by (16),
^V=\\^^^^\i.Adudvdw,
where rfw, rfv, and dtv are positive increments, and where c is infini-
tesimal with du, dv, and dw. The term e du dv dw may be neglected,
M has been explained several times (§ 128). The product
(17) rfK = P^-^"»'^)
du dv dw
\D{Uy V, w)
is the principal part of the infinitesimal AF, and is called the element
of volume in the system of curvilinear coordinates (u, v, w).
Let d** be the square of the linear element in the same system of
coordinates. Then, from (6),
whence, squaring and adding, we find
X ^Hxdu^-¥Utdf)^-\-lUdw^-\-2F^dvdw+2F^dudw+2F,,dudv,
VII. J 146] INTKODUCTION CHANGE OF VARIABLES S05
the notation employed being
(19)
(".-^©■' "sm: «..sm:
'Bxy
F»
^ bu cv
where tlie symbol S means, as usual, that x is to be replaeed by y
and « successively and the resulting terms then added.
The formula fcr dV is easily deduced from this formula for d^.
For, squaring the functional determinant by the usual rule, we find
Af,
whence the element of volume is equal to "VMdu dv dw.
Let us consider in particular the very important caae in whieh
the coordinate surfaces (u), {v)y (ta) form a triply orthogonal Bjtt«m,
that is to say, in which the three surfaces which pass through any
point in space intersect in pairs at right angles. The tnngentt to
the three curves in which the surfaces intersect in pairs form a tri-
rectangular trihedron. It follows that we must have F} sa 0, F, ■■ 0,
F, = ; and these conditions are also sufficient The formubs for
rfl' and ds^ then take the simple forms
dx
dj.
dx
t
du
du
du
//l
F,
Ft
dx
hi
dv
dz
dv
dv
=
Ft
//.
Fi
dx
d_y
dx
Ft
^1
B.
dw
dw
dw
du dv dw.
(20) ds^ = Hi du' 4- //, dv* + //, dw*, dV = >///,//,//.
These formulae may also be derived from certain considerations of
infinitesimal geometry. Let us suppose du^ dv, and dw very small,
and let us substitute in place of the small subregion defined ftbore a
small parallelopiped with plane faoes. Neglecting infinit esim a l s of
higher order, the three adjacent edges of the parallelopiped may be
taken to be yJlT^du, y^/Tf^dv, and y/W^dw, respectively. Tbe for-
mulae (20) express the fact that the linear element and the elemeot of
volume are equal to the diagon al and the volume of this parallelo-
piped, respectively. The area v/i/j //, du dv of one of the faoes repre-
sents in a similar manner the element of area of the surface (w).
As an example consider the transformation to polar coordinates
(21) x = psindco8</>, y = p8intf8in4, s«peos#,
306
MULTIPLE INTEGRALS
[VII, § 146
where p denotes the distance of the point M(Xf y, z) from the origin,
B the angle between OM and the positive ;;; axis, and <^ the angle
which the projection of OM on the xij plane makes with the positive
X axis. In order to reach all points in space, it is sufficient to let p
vary from zero to 4- oo, ^ from zero to tt, and <^ from zero to 27r.
From (21) we find
(22) ds''
whence
dp^ -^ p^dO'' ^ p'^ sm''ed4>'^,
(23) dV = p"^ sine dp ddd<i>.
These formulae may be derived without any calculation, however.
The three families of surfaces (p), (d), (<^) are concentric spheres
about the origin, cones of revolution
about the z axis with their vertices
at the origin, and planes through
the z axis, respectively. These
surfaces evidently form a triply
orthogonal system, and the dimen-
sions of the elementary subregion
are seen from the figure to be dp,
p d$, p sin e d(f> ; the formulae (22)
and (23) now follow immediately.
To calculate in terms of the va-
riables p, 6, and </> a triple integral
extended throughout a region bounded by a closed surface S, which
contains the origin and which is met in at most one point by a radius
vector through the origin, p should be allowed to vary from zero to R,
where R =/($, <f>) is the equation of the surface ; $ from zero to
Fio. 34
tt:
and <^ from zero to 27r. For example, the volume of such a surface is
X2.T ^n pR
d<t> j del p^ sine dp.
The first integration can always be performed, and we may write
OocMional use is made of cylindrical coordinates r, o), and z defined
by the equations x = r cos «, y = r sin cu, « = z. It is evident that
dV =si r dit) dr dz ,
VII. §147] INTRODUCTION CHANGE OP YARUBLES ttfj
147. Elliptic coii^dinatet. The mutmom reprsMBled by ut« e4uatK]a
X* V* «•
X-o X-6 X-c
%rhere X Ib a rariable parameter and a > 6 > c> 0, fom a teailj o(
conies. Through every point in space there pa« three anrfaees of this faadljr,—^
an ellipsoid, a parted hyperboloid, and an unparted hjperbolold. For the eq«»>
tion (24) always has one root Xi which liee betwee n b and c, another rooi X«
between a and 6, and a third root Xt greater than a. Theee three roou X|, X«, Xg
are called ihe elliptic coordinaUa of the point whoee rectangular frfrftrdinaif are
(z, y, z). Any two surfaces of the family int«reeet at right angles: if X he given
the values Xi and X3, for instance, in (24), and the rseoltlng eqoationa be snb-
tracted, a division by Xi — Xt glyes
^ ' (Xi-a)(X,-a) (Xi-6)(X,-6) (Xj - c)(X, - c)
which shows that the two surfaces (Xi) and (X9) are orthogcmal.
In order to obtain z, y, and z as functions of Xi , Xf , Xa , we maj note that the
relation
(X _ a)(X - 6)(X - c) - x^{\ - b){\ _ c) - y«(X - c)(X - a) - f«(X - «)(X - b)
= (X - X|)(X - X,)(X - X.)
is identically satisfied. Setting X = a, X = 6, X = e, i ucn ee i irely, in tide eqQ»-
tion, we obtain the values
r,,_ (X«-a)(a~X,)(a~X,)
I (a-b)(a-c) •
(X. - 6)(X, - b){b - XQ
(26)
V (a-6)(6-c)
^ ^ (X, - c)( X,-c)<Xi~c)
(a - c)(6 - c)
whence, taking the logarithmic deriTatiree,
S \X, - o X, - « X, - «/
,/ dX, (IX. (ft, \
*-iii:r^e+i:r^e+xr^«r
Forming the sum of the squares, the terms In (iX,tfX«, dXfAa, AeA| »■* dto
appear by means of (26) and similar relaUons. H^ne* the ooeiBeioiil of tfX, li
^' " 4 L(xr^« "*" (xr=^ "" ^ - «>• J '
or, replacing x, y, t by their valnee and simplifying,
(X« - XO(Xt - X|)
(27) Ml ^2
4 (Xi - a)(X, - 6)(X, - e)
$08 MULTIPLE INTEGRALS [VII, §148
eoeflleients lit and Jf| of <ix| and dxj, respectively, may be obtained from
ion by cyclic permutation of the letters. The element of volume is
thtref ore VMiMt Mt d\x dX, dX. .
141. Dirichlet't integrals. Consider the triple integral
xPy««'"(l — X — y — z)'dxdydz
fff'
throaghout the interior of the tetrahedron formed by the four planes
ssO, ir = 0, « = 0, x4-y + «=l. Let us set
x + y-\-z = i, y + z = ^v, z = ^r]^,
where (, If, f are three new variables. These formulae may be written in the form
y -\- z ^ z
x + y + z y + z
and the inverse transformation is
x = ^{i-v), y = ^v{i-^), z = hj:.
When X, y, and z are all positive and x + y + zis less than unity, ^, rj, and f all
lie between zero and unity. Conversely, if ^, ?;, and f all lie between zero and
unity, x, y, and z are all positive and x-^y + z is less than unity. The tetra-
hedron therefore goes over into a cube.
In order to calculate the functional determinant, let us introduce the auxiliary
transformation JT = f , r = fi;, Z = ^r^f , which gives x = X-Y, y = Y-Zy
t = Z. Hence the functional determinant has the value
D(x,y, z) ^ D(x, y, z) J)(X, F, Z) ^
D(^7,, f) i)(X, r, Z)" D(|, ^, f) ^'''
and the given triple integral becomes
The integrand le the product of a function of {, a function of rj, and a func-
tion of f. Hence the triple integral may be written in the form
or, introducing r functions (see (33), p. 280),
Tip + q-^r-^ B)r{i +i) ^^ r(g -f r -t- 2)r(p +1) r(r + i)r(g4-i)
r(p + y + r + « + 4) r(p + g + r + 3) ^ r(94-r + 2) '
CanotUnf the common factors, the value of the given triple integral is finally
found to he
(88) r(p4-l)r(g4-l)r(r.fl)r(a+l)
r(p + 5 + r + « + 4)
VII. §149] INTKODUCTION CHAMGE OF VAKIABLES 809
149. Green'i thwutm.* A formula entirely MuOogoiu lo (lo^, | IM, m»y te
derived for triple integralB. Let us arat coiulder a etoaed torfiM 8 wkUk to
met in at most two poinu by a parallel to the < axii, and a funeUoo A(s, y, g)
which, together with dH/^x, Is continuous thrauglkout the iniarior ol tbianrteoa.
All the points of the surface 8 project into poinu of a ragloD il ollbe ay r^^
which iH bounded by a closed contour C. To every point of A »— Mt C eon*-
spond two pointo of 8 whose coordinates are <i = ^i(z, y) and sg a ^(x« y).
The surface S is thuM divided into two distinct fxiriions ^i aod iBg. We ««*i ^«
suppose that Zi is less than Zt.
Let us now consider the triple integral
///
— ozdyds
£z
taken throughout the region bounded by the closed surface 5. A .-^ ,,, ,,
tion may be performed with regard to t between the limita Mt and ft (f 144),
which gives K(x, y, zt) - K{x, y, zi). The given triple Integral to thwifwi
equal to the double integral
//[«(x, y, Zt) - «(x, y, tOJdedy
over the region A. But the double integral ffR(x, y, c«)dzdy to equal to the
surface integral (§ 13o)
ff R{x,y,z)dx(fy
taken over the upper side of the surface St . Likewise, the
R{x, y, zi) with its sign changed is the surface integral
ff R(x,y,t)dxd^
8i . Adding these tw
taken over the lower side of 8i . Adding these two Integrato, we may
where the surface integral is to be extended over the whole mimriar of the
face S.
By the methods already used several times in similar caaee tl
be extended to the case of a region bounded hy ^warho^oiMaj
Again, permuting the letters z, y, and t, we obtain the aaalogoat
///
///
~dxdvdz=ff P(x, y, f)d|rdi,
^^dzdy<U=ff Q(x, y, s)dt4a.
dy J JfS)
• Occasionally called Osfro^rodciby'* theorem. The t h eo t e w of f IS to i
called Iiiema7m'.H theorem. But the title Oreen*» CAsorem to SMwe etoariy
and seems to be the more fitting. See Mnqf, dtt Mmtk. IRss., II, A, 7. h aad e.—
Trams.
^^ MULTIPLE INTEGRALS [VII, §150
Adding thm& three formula, we finally find the general Green's theorem for
triple integrals :
(2D)
= ff P(x, y, z)dydz + Q{x, y, z)dzdx + 12(x, y, 2:)(icdy,
J J{5)
when the surface integrals are to be taken, as before, over the exterior of the
bounding surface.
If, for example, we set P = x, Q = R = or Q = y, P = R = or R = z,
P = Q = , it is evident that the volume of the solid bounded by S is equal to
Any one of the surface integrals
(290
IL"''^'' /i/"'^' /^^'"'^
150. Multiple integrals. The purely analytical definitions which have been
gi?en for double and triple integrals may be extended to any number of vari-
ables. We shall restrict ourselves to a sketch of the general process.
Let zi , Xt , • • • , x„ be n independent variables. We shall say for brevity
that a system of values x}, x§, • • • , xj of these variables represents a point in
space of n dimensions. Any equation F(xi, X2 , • • • , x„) = 0, whose first member
is a continuous function, will be said to represent a surface ; and if F is of the
first degree, the equation will be said to represent a plane. Let us consider the
totality of all points whose codrdinates satisfy certain inequalities of the form
(80) ^.(xi,««, •••,x„)<0, i = l, 2, ..., A:.
We shall say that the totality of these points forms a domain D in space of n
dimensions. If for all the points of this domain the absolute value of each of
the cotedinates x, is less than a fixed number, we shall say that the domain B is
finite. If the inequalities which define D are of the form
(81) «;<«i<xi, x;<x,<x5, ..., x;<x„<xl,
we shall call the domain a priamoid, and we shall say that the n positive quan-
tities x} - a^ are the dimensions of this prismoid. Finally, we shall say that a
point of the domain D lies on the frontier of the domain if at least one of the
functions ft in (80) vanishes at that point.
Now let D be a finite domain, and let /(xi , Xa , • • • , Xn) be a function which
b continuous in that domain. Suppose D divided into subdomains by planes
pAfallei to the planes x, = (f = 1, 2, •• • , n), and consider any one of the pris-
moida determined by these planes which lies entirely inside the domain D.
JM Asi , Axtt " - 1 Ax» be the dimensions of this prismoid, and let fi , fa , • • • , t,
bt tba coordinates of some point of the prismoid. Then the sum
(tl) fl = 2/({„ {,, . . ., („) AxiAx,. . . Ax«,
formad for all the prlsmoids which lie entirely inside the domain D, approaches
a limit / as the number of the prlsmoids is increased indefinitely in such a way
VII, §150] INTRODUCTION CHANGK OF VAUIABLES 311
that all of the dimenaiooi of each of ibem np\n
limit / the n-tuple integral of/(Xi, 2«, • • •, z.) ta.
denote it by the symbol
/=// • //(xi. X,. .. .. x,)d£,(ljr.
The evaluation of an n-tuple integral may be reduced to the etaJoalloci ci
n successive simple integrals. In order to show this In general, we need only
show that if it is true for an (n - l)-tuple integral, it will aleo be Ime for an
»-tuple integral. For this purpose let us consider any point (Xi, St«* *-• <■)
of D. Discarding the variable x« for the moment, the point (Zi , Zt t "*•'«- 1) c^
dently describes a domain 1/ in space of (a - 1) dlmendona. We shall sappoM
that to any point (xi, xt, • •, z«-i) inHide of IX there oorreepond Juet two
points on the frontier of D, whoee coordinates are (zi, Xt, • • •, z..! ; z^J^ and
(xi , za , • • • , Xm-1 ; z^/^), where the coordinates z^^ and z^" are oontinuoue fan6>
tions of the n - 1 variables Zi , z^ , • • • , z.-i inside the domain IX. If this eeo-
dition were not satisfied, we should divide the domain D into domains i
that the condition would be met by each of the partial domains. Let
consider the columu of prismoids of the domain D which correspond to the
same point (zi, z^, • • •, z«-i). It is eaey to show, as we did in the similar eaae
treated in § 124, that the part of S which arlaet from thia column of prtaHwida ii
AziAxt ••Az,_irr '/(zi, %,..., jgdz, + €j.
where |e| may be made smaller than any positive number wbaterer by cIkk»>
ing the quantities Az, sufficiently smalL If we now set
(88) ♦(zi,z,,...,z,_,)=j^,^/lzi,a^,...,*jdz..
it is clear that the integral I will be equal to the limit of the
2*(zi, z«, •••, z,_i)AzjAZf.Az,_i,
that is, to the (n - l)-tuple integral
(84) 1 = JJJ •.J*(zi,z,,. •,z._i)dzi...dj|,-i,
in the domain IT. The law having been ■oppoeed to bold for an (• - IHaple
integral, it is evident, by mathematical indadkm, that It bokla in geoeraL
We might have proceeded difterenUy. Consider the loulity of points
(xu Xt, . . •, Xn) for which the coordinate z, has a fixed valne. Tbeo the
point (zi, zi, . • •, Xn-i) describes a domain i in apnea of (n - 1) dIavMinm,
and it is easy to show that the Wrtuple integral / ia alio equal to the ■ipiaaliia
■ (86) I^f^eiz.)dx..
where 0{Xn) is the (n - l)-tuple integral ///• • f/dtt • • • da. - 1 ezlMided throogb.
out the domain «. Whatever be the method of carrying oat Uie proesM, the Umlta
for the various integrations depend upon the naiiire of Ibe donate A ■■*
312 MULTIPLE INTEGRALS [VII, §160
Tary to general for different orders of integration. An exception exista in case
D is a priamoid defined by inequalities of the form
a?^«i<Xi, •., xJ<x,<X., ....
The multiple Integral ia then of the form
and the oitler in which the integrations are performed may be permuted in any
way whaterer without altering the limits which correspond to each of the
variables.
The formula for change of variables also may be extended to n-tuple integrals.
Let
(36) x, = 0,(xi',a^, •••,x;), i = l, 2, ..-, n,
be formuliBof transformation which establish a one-to-one correspondence between
the points (x'l, Xa » • • • , Xn) of a domain 1/ and the points (xi, Xa, • • • , x„) of a
domain D. Then we shall have
ff"'f ^(^i'^'*"'^«)^i""^»
(87)
J J J(L
F{<f>i » • • • » 0«)
-D(0i, •••,0«)
I>(xi, ••«, x;)
dxi- • ■ dXn .
The proof is similar to that given in analogous cases above. A sketch of the
argument is all that we shall attempt here.
1) If (37) holds for each of two transformations, it also holds for the trans-
formation obtained by carrying out the two in succession.
2) Any change of variables may be obtained by combining two transforma-
tions of the following types :
(88) «i = xi, x, = x$, •.., x„_i = x;_i, x„ = 0„(x{, x^, ..-, x;),
(89) «i = fi(xi,..-,x;), ..., Xn-i = i/'„-i(xl, .-.jx;), x„ = xA.
8) The formula (37) holds for a transformation of the type (38), since the
given yi-tuple integral may be written in the form (34). It also holds for any
transformation of the form (39), by the second form (35) in which the multiple
totegral may be written. These conclusions are based on the assumption that
(87) holds for an (n — l)-tuple integral. The usual reasoning by mathematical
toduction establishes the formula in general.
As an example let us try to evaluate the definite integral
^ =//• • -/ac^xj' . . . x;-(l - xi - xa Xn)^dxidX2. . .dx„,
where ai, at, ■ • , or., /3 are certain positive constants, and the integral is to be
CTt ended throughout the domain D defined by the inequalities
0<x,, 0<Zs, ..., 0<x,., Xi + X2 + .-. + »,<l.
The transformation
Vll, §161] TOTAL DIFFEUENTlAiJi $||
carries I> into a new domam if ueiineu by loe ineqiuuiuet
0<(i<l, 0^|,<1, ..., 0<|.^1,
and it is easy to show as in f 148 that the Tains of th« •■^■w»Hi^af rtstwlnm to
^git«tt "s a^i) _>•-!>■ -t .
iAJn^ (tt •••, Km)
The new integrand is therefore of the form
and the given integral may be expressed, as before, in terms of r fnoctioBS i
(40) I = ^'^^ 4-l)r(a , H-l)...r(a,.H)r(^^l)
r(ai + a, + • • • + a. + /I + m + 1)
n. INTEGKATION OF TOTAL DIFFERENTIALS
151. General method. Let /'(x, y) aiid Q(x, y) be two functions of
the two independent variables x aiid y. Then the expreeeioo
Pete -{- Qdy
is not in general the total differential of a single function of the two
variables x and y. For we have seen that the equation
(41) du^Pdx^-Qdy
is equivalent to the two distinct equations
(42) t='P(^.y). |^=«(''y)-
Differentiating the first of these equations with reepeot to y and tlie
second with respect to x, it appears that m(z, y) must satisfy eaoh
of the equations
c^u ^ dP(x,y) Bl^u ^ dQ(x,y)
dxdy" dy ' dydx"^ dx
A necessary condition that a function «(«, y) should exist which
satisfies these requirements is that the equation
(''3) ay to
should be identically satisfied.
This condition is also st^ffleieni. For there exist an in6nit»
number of functions w(x, y) for which the first of equations (43)
is satisfied. All these functions are given by the formula
u^j Pix,y)dX'^Y,
814 MULTIPLE INTEGRALS [VII, §151
where oiiiq is an arbitrary constant and Y is an arbitrary function of y.
In order that this function u{x^ y) should satisfy the equation (41),
it is necessary and suflScient that its partial derivative with respect
to X should be equal to Q(Xj y), that is, that the equation
should be satisfied. But by the assumed relation (43) we have
whence the preceding relation reduces to
dY
^ = Q(^o,y).
The right-hand side of this equation is independent of x. Hence
there are an infinite number of functions of y which satisfy the
equation, and they are all given by the formula
Y=\ Q(x,,y)dy^C,
where y© is an arbitrary value of y, and C is an arbitrary constant.
It follows that there are an infinite number of functions u{x, y)
which satisfy the equation (41). They are all given by the formula
(44) u=\ P(x, y)dx-h f Q(xo, y)dy + C,
and differ from each other only by the additive constant C.
Consider, for example, the pair of functions
a; + my ^ _ y - ma;
x' + y'' ^-^M^'
which satisfy the condition (43). Setting a^o = and y^ = 1, the
formula for u gives
Jo x^ + y^ J^ y »
whence, performing the indicated integrations, we find
= 2 P*^»(** + y*)]* + ^'^ r^c tan -T 4- log y 4- e,
U =s
or, simplifying,
1
tt = 2 lo«(»* ■fy*) + marctan--hC,
VII, 5 151] TOTAL DIFFERENTIALS $16
The preceding method may be extended to any number of inde-
pendent variables. We shall give the reasoning for tbree Tariablet.
Let Py Qj and R be three functions of x^ y^ and z. Then the total
differential equation
(46) du = luU -j- Udi/ -}- It dz
is equivalent to the three distinct equatiouw
Calculating the three derivatives d^u/dxdy^ dl*u/dybUfd^u/dzbx in
two different ways, we find the three following equationi at naoea-
sary conditions for the existence of the function m;
(47) ^ = ^, £^ = ^, i^^^.
dy dx* Bk dy* dx du'
Conversely, let us suppose these equations satisfied. Then, bv tiie
first, there exist an infinite number of functions u{Xy y, t) whose
partial derivatives with respect to x and y are equal to P and Q,
respectively, and they are all given by the formula
t*=/ P{x,y,z)dx^\ Q{Xf,,y,z)dy'k'Z,
where Z denotes an arbitrary function of z. In order that the derira-
tive du/dz should be equal to i?, it is necessary and sufficient that
the equation
should be satisfied. Making use of the relations (47), whieh wet*
assumed to hold, this condition reduces to the equation
dZ
R(x, y, z) - /e(Xo, y, z) -f R(x^, y, z) - R(x., y., «) + ^ = i?(x, y, s) ,
It follows that an infinite number of functions u(x, y, z) exist
which satisfy the equation (46). They are all giren by the fonnula
(48)
u= f P(x, y,z)dx-h f Qix^y y. r^ du + ( R(x., y.. z^ dx + r.
where Xq, i/,,, z^, are three arbitrary numerical values, and r i> ;in
arbitrary constant.
310 MULTIPLE INTEGRALS [VU,§162
152. The Integral /^'J^Pdx-fQdy. The same subject may be
treated from a different "point of view, which gives deeper insight
into the question and leads to new results. Let P(x, y) and Q{xy y)
be two functions which, together with their first derivatives, ai-e
continuous in a region A bounded by a single closed contour C.
It may happen that the region A embraces the whole plane, in
which case tlie contour C would be supposed to have receded to
infinity. The line integral
/
Pdx + Qdy
taken along any path D which lies in A will depend in general upon
the path of integration. Let us first try to find the conditions under
which this integral depends only upon the coordinates of the extremi-
ties (jTg, y^ and (ajj, y^) of the path. Let M and N be any two points
of region Ay and let L and V be any two paths which connect these
two points without intersecting each other between the extremities.
Taken together they form a closed contour. In order that the values
of the line integral taken along these two paths L and V should be
equal, it is evidently necessary and sufficient that the integral taken
around the closed contour formed by the two curves, proceeding
always in the same sense, should be zero. Hence the question at
issue is exactly equivalent to the following : What are the conditions
under which the line integral
I
Pdx-{-Qdy
taken around any closed contour whatever which lies in the region A
should vanish ?
The answer to this question is an immediate result of Green's
theorem :
(49) i^-^^<^^y=Jfl^i-'iy^y>
where C is any closed contour which lies in .4, and where the double
integral is to be extended over the whole interior of C. It is clear
that if the functions P and Q satisfy the equation
the line integral on the left will always vanish. This condition is
necessary. For, if cP/dy - dQ/dx were not identically zero
VII, §152] TOTAL DIFFEUEN'TrAW gl7
in the region .4, since it is a continuuu^ luuction, it would turelj be
possible to find a region a so small that its sign would be oonstant
inside of a. But in that case the line integral taken around the
boundary of a would not be zero, by (49).
If the condition (43*) is identically satisfied, the Talnei of the
integral taken along two paths L and /.' between the same two
points M and N are equal provided the two paths do not intarteei
between M and N. It is easy to see that the same thing is true
even when the two paths intersect any number of times between Mt
and ^V. For in that case it would be necessary only to oompare
the values of the integral taken along the paths L and L* with its
value taken along a third path L'\ which intersects neither of the
preceding except at M and N.
Let us now suppose that one of the extremities of the path of
integration is a fixed point (xo, y^y while the other extremity is a
variable point (x, y) oi A. Then the integral
I Pdx + Qdy
<'•• »•>
taken along an arbitrary path depends only upon the oo6rdinates
(x, y) of the vaiiable extremity. The partial derivatiTee of this
function are precisely P(x, y) and Q(x, y). For example, we have
for we may suppose that the path of integration goes from (x^, f^)
to (x, y),.and then from (x, y) to (x + Ax, y) along a line parallel to
the X axis, along which dy = 0. Applying the law of the mean, we
may write
Ax
Taking the limit when Ax approaches »ro, this gives P^»P,
Similarly, F, = Q. The line integral F(x, y), therefore, satisfies the
total differential equation (41), and the general integral of this
equation is given by adding to F(x, y) an arbitrary constant
This new formula is more general than the formula (44) in that
the path of integration is still arbitrary. It is easy to deduce (44)
from the new form. To avoid ambiguity, let (x«, y.) and (x,. y,) be
the co5rdinates of the two extremities, and let the path of int4^r»-
tion be the two straight lines x = x^, y = yi. Along the former,
818 MULTIPLE INTEGRALS [Vll, §153
X =1 Xo» dx = Of and y varies from ijo to yi. Along the second,
y = yi, rfy = 0, and X varies from Xq to a^i. Hence the integral (50)
is equal to
J' 'Q(xoyy)dy+f P(x,yi)dx,
which differs from (44) only in notation.
But it might be more advantageous to consider another path of
integration. Let x = /(^), y = <f>(t) be the equations of a curve
joining (xo, y©) and (xi, yi), and let t be supposed to vary con-
tinuously from to to ti as the point (x, y) describes the curve
between its two extremities. Then we shall have
r '"'^Pdx + Qdy^f \p(x, y)f'(t) + Q{x, y) ^\t)-] dt,
where there remains but a single quadrature. If the path be
a straight line, for example, we should set x = Xq-}- t(xi — Xq)j
y = yo + t(yi — yo), and we should let t vary from to 1.
Conversely, if a particular integral ^(x, y) of the equation (41)
be known, the line integral is given by the formula
I Pdx + Qdy=:^(x,y)-^(xQ,yQ),
which is analogous to the equation (6) of Chapter IV.
153. Periods. More general cases may be investigated. In the
first place, Green's theorem applies to regions bounded by several
contours. Let us consider for definiteness a region A bounded by
an exterior contour C and two contours C and
C" which lie inside the first (Fig. 35). Let P
and Q be two functions which, together with
their first derivatives, are continuous in this
region. (The regions inside the contours C
and C" should not be considered as parts of
the region A^ and no hypothesis whatever is
made regarding P and Q inside these regions.)
Let the contours C and C" be joined to the contour C by trans-
▼eraals ab and ed. We thus obtain a closed contour abmcdndcpbaqay
or r, which may be described at one stroke. Applying Green's
theorem to the region bounded by this contour, the line integrals
VU,J153] TOTAL DIFFERENTIALS 810
whicli arise from the traDBversaU ab and cd eanoel out, tioM » ^h
of them is described twioe iu opposite direotiona. It follows that
where the line integral is to be taken along the whole boondarj of
the region A, Le. along the three contours C, C, and C'\ in tho mmum
indicated by the arrows, respectively, theso being tuoh that tha
region A always lies on the left
If the functions P and Q satisfy the relation dQ/dx » BP/^ in
the region .1, the double integral vanishea, and we may write the
resulting relation in the form
(51) r Pdx + Qdi/= f Pdx-i- Qdy -f- / Prfx + Wy,
JiC) J{V) J{C"}
where each of the line integrals is to be taken in the aenae dcaig-
uated above.
Let us now return to the region A bounded by a single eontoor
C, and let P and Q be two functions which satisfy the eqoatioo
dP/dy = dQ/dxy and which, together with their first deriTatiTea, are
continuous except at a finite number
of points of Af at which at least one of
the functions P or Q is discontinuous.
We shall suppose for definiteness that
there are three points of discontinuity
a, by c in .4. Let us surround each of
these points by a small circle, and then
join each of these circles to the contour
C by a cross cut (Fig. 36). Then the
integral jPdx-\-Qdy taken from a fixed
point (xo, yo) to a variable point (ar, y) y,Q. as
along a curve which does not croes any
of these cuts has a definite value at every point For the ooaUwr C,
the circles and the cuts form a single contour which may be deeeribed
at one stroke, just as in the case disouased aboTO. We iliall eall
such a path direct, and shall denote the ral oe of t he line intef^
taken along it from 3fo(j-o, y*) to3f(a:, y) by F{x, y).
We shall call the path composed of the straight line from U^ u>
a point a\ whose distance from a is infinites im al, the eironMfwsMa
of the circle of ra<liu8 aa' about a, and the straight line a*M%t a l»^
circuit. The line integral jPdx + Qdy taken along a loop-emiil
820 MULTIPLE INTEGRALS [VII, §153
reduoes to the line integral taken along the circumference of the
circle. This latter integral is not zero, in general, if one of the
functions P or Q is infinite at the point a, but it is independent of
the radius of the circle. It is a certain constant ± A, the double
sign corresponding to the two senses in which the circumference
may be described. Similarly, we shall denote by ± B and ± C the
values of the integral taken along loop-circuits drawn about the two
singular points h and c, respectively.
Any path whatever joining Mq and M may now be reduced to a
combination of loop-circuits followed by a direct path from Mq to M,
For example, the path MomdefM may be reduced to a combination
of the paths M^^mdMo) ModeMo, MoefMoy and M^fM. The path
M^mdM^ may then be reduced to a loop-circuit about the singular
point a, and similarly for the other two. Finally, the path M^fM
is equivalent to a direct path. It follows that, whatever be the path
of integration, the value of the line integral will be of the form
(52) F(x, y) = Fix, y) -f mA -f tiB + ^C,
where m, n, and p may be any positive or negative integers. The
quantities A, B, C are called the periods of the line integral. That
integral is evidently a function of the variables x and y which
admits of an infinite number of different determinations, and the
origin of this indetermination is apparent.
Remark, The function F{x, y) is a definitely defined function
in the whole region A when the cuts aa, bfi, cy have been traced.
But it should be noticed that the difference F(m) - F(rriJ) between
the values of the function at two points m and m' which lie on
opposite sides of a cut does not necessarily vanish. For we have
' + / + I ,
M^ Jm Jmf
which may be written
But/Ji«zero; hence
F(^-F(^= A.
It follows that the difference F{rn) - F{m^ is constant and equal
to A all along aa. The analogous proposition holds for each of
the cuU.
VII, JIM] TOTAL DUFERENTIALS «fl
Example. T^^** Ti»'« integral
4/(1.1
xdy — ydx
'0.0, x' + y*
has a single critical point, the origin. In ortler to find tlw
spending period, let us integrate along the circle jt'-I-^m^.
Along this circle we have
x = pcosw, y = psin«u, xdy - ydx = p'^dm,
whence the period is equal to ^*V« = 2ir. It is easy to Terify
this, for \hiy intf^trrand is the total differential of arotmny/z.
154. Common roots of two eqaations. Let X and T be two fanctloiMof llw
variables z and y which, together with their first partial deriTatlves, a
tinuous in a region A bounded by a single eloeed contour C. Then the
8ion {XdY - YdX)/{X^ + Y*) satisfiee the oondiUon of integrabUitj, for U to
the derivative of arc tan Y/Z. Hence the line Integral
(53) , ZdT-TdX
I
taken along the contour C in the poeitive sense vaotohea
cients of dx and dy in the integrand remain continoous Insida C, La if ths two
curves X = 0, y = have no common point inside that oootomr. But tf tlMst
two curves have a certain number of common points a, ft, c, • • • TnsMt C,
of the integral will be equal to the sum of the values of ths i
along the circumferences of small circles described about tbs potots a, A^ e, • • • m
centers. Let (a, /3) be the coordinates of one of the comiiKNl poiala Wa skall
suppose that the functional determinant D(X, Y)/D{z, y) to not asro. La. llMi
the two curves X = and F = are not Ungent at the point. Tbsa it to poa->
sible to draw about the point (a, fi) as center a circle c whose radios to so saaU
that the point (X, Y) describes a small plane region about ths point (0, 0|
which is bounded by a contour y and whicb oo r rssponds poiai for poteS to tiM
circle c (§§ 25 and 127).
As the point (x, y) describes the circumfSrenos of tka olroto e la ths
sense, the point (X, Y) describes the contour > In the positive or la tbs
sense, according as the sign of the functional datanaiBaDt iarida tbs etaela s to
positive or negative. But the deflnita Integral sloag the drcnadmrnmoi e to
equal to the change in arc tan Y/X in one revolution, that to, ±
reasoning for all of the roots shows that
/
<"> L^^^="<^-'>-
where P denotes the number of poinds common to ths two eorres at whfcb
D(A', Y)/D{x, y) is posiUve, and .Y ths number of ooonaon polato al whteb tbs
determinant is negative.
822 MULTIPLE INTEGRALS [VII, §155
Hm definite integral on the left is also equal to the variation in arc tan Y/X
fat going aruuud c, that is, to the index of the function YJX as the point (x, y)
dMoribet the contour C If the functions X and Y are polynomials, and if the
oontoor C is composed of a finite number of arcs of unicursal curves, we are led
to calculate the index of one or more rational functions, which involves only
elementary operations (§ 77). Moreover, whatever be the functions X and Y^
we can always evaluate the definite integral (54) approximately, with an error
lees than r, which is all that is necessary, since the right-hand side is always a
maltiple of In.
The fonnula (64) does not give the exact number of points common to the
two cur>'e« unless the functional determinant has a constant sign inside of C
Picard*8 recent work has completed the results of this investigation.*
199. Gmieralixation of the preceding. The results of the preceding paragraphs
may be extended without essential alteration to line integrals in space. Let P,
Q, and /? be three functions which, together with their first partial derivatives,
are continuous in a region (E) of space bounded by a single closed surface iS.
Let OS seek first to determine the conditions under which the line integral
(66) Cr= / P(to + Qd?/ + -Bd2
depends only upon the extremities (xo, yo, z^ and (x, y, z) of the path of inte-
gration. This amounts to inquiring under what conditions the same integral
ranishes when taken along any closed path r. But by Stokes' theorem (§136)
the above line integral is equal to the surface integral
//(S-5)"-(s-'>-(f-S>'
extended over a surface S which is bounded by the contour r. In order that
this surface integral should be zero, it is evidently necessary and sufficient that
the equations
(6«) sP_aQ aQ_aB ^_5^
ay ax * Zz~ dy' dx ~ dz
ihoald be satisfied. If these conditions are satisfied, U" is a function of the vari-
ables X, y, and z whose total differential is P dx + Q dy + iJ dz, and which is single
valued in the region {E). In order to find the value of U at any point, the path
of integration may be chosen arbitrarily.
If the functions P, Q,and R satisfy the equations (66), but at least one of
them becomes Infinite at all the pomts of one or more curves in (JK), results
analogous to those of § 153 may be derived.
If, for example, one of the functions P, Q, R becomes infinite at all the points
of a closed curve 7, the integral U will admit a period equal to the value of the
line integral Uken along a closed contour which pierces once and only once a
•orfaoe r bounded by 7.
We may also consider questions relating to surface integrals which are exactly
analogous to the questions proposed above for line integrals. Let ^, B, and C
be three functions which, together with their first partial derivatives, are
•TraU4 d'AnatyBe, \o\.l\.
Vll.fWfi] TOTAL D1FFERKNTIAL8
contlnuouA in a region(JSr)of >p«oeboaiidMlb7adjifUeloMdMrteM& LkZ
be a surface inside of (E) bounded by a oootour r of any form whaieTtr.
the surface integral
(67) ^=rr Ad^dg-^Bdxdx-i-Cdzdv
J J(2)
depends in general upon the surface Z as well as upon tha eonloar r. la
that the integral should depend only upon r, it la eridaDtly naosHary aad
cient that its value when taken over any closed stufiM In (JT) sliottld vaaWi.
Green's theorem (§ 140) gives at once the conditions under wlUeh this Is tma.
For we know that the given double integral extended orer any clowd nrfaea ii
equal to the triple integral
///(g*5'*f)
dx<^dg
extended throughout the region bounded by the wirffww In ofder lluii tiili litlar
integral should vanish for any region inside (i?), it is evidently necessary that Um
functions Ay B, and C should satisfy the equation
This condition is also sufBcient.
Stokes' theorem affords an easy verlficatioD of this fact. For If A^ B, and C
are three functions which satisfy the equation (68), It is always possible to deter-
mine in an infinite number of ways three other functions P, Q, and R
(69) ««_£« = ^. »l-»Ji^B. ?«-»f=C.
^ ' By dz H dz to «y
In the first place, if these equations admit solutions, they admit an
number, for they remain unchanged if P, Q, and R be replaeed by
tx ty H
respectively, where X Is an arbitrary function of x, y, and t. Again,
R=Q, the first two of equations (60) gire
P= C B(x, y, z) dz + 0(x, y) , Q = - f 'yl(x, y, «)ds + f (a. f).
where <p{x, y) and v('(x, y) are arbitrary functions of z and y.
values in the last of equations (50), we find
a-g.cB.,.«,
or, making use of (58)
One of the functions or ^ may still be chosen at
The functions P, Q, and R having been deCermioed, the sortMa latsfral, kj
Stokes' theorem, is equal to the Une integral /tn^^*^ Q^ + **• •**^'*^
evidently depends only upon the oontoor V.
SSi MULTIPLE INTEGRALS [VII, Exs.
EXERCISES
1. Find the valae of the triple integral
r r r[6(x - yy + Sae - ia^]dxdydz
Vt^ m^'^ throughout the region of space defined by the inequalities
x« + y«-a«<0, xa + y2_|.22_ 2a2<0.
[Licence^ Montpellier, 1896.]
2. Find the area of the surface
^ ^ ^ a2x2 + 62y2
and the yolume of the solid bounded by the same surface.
3. Inrestigate the properties of the function
F{X, Y, Z) = f'dx ( dy( /(x, y, z)dz
eootidered as a function of X, F, and Z. Generalize the results of § 125.
4. Find the volume of the portion of the solid bounded by the surface
(x2 + y2 + z2)8 _ Z(j%xyz
which lies in the first octant.
6. Reduce to a simple integral the multiple integral
//' * '/*''^»' ' ■ ' ^«"^(^i + X2 + • . . + x„)(toi(te2. . -dxn
extended throughout the domain D defined by the inequalities
O^xi, 0<x,, •.., 0<x„, xi + X2 + --.+x„<a.
[Proceed m in § 148.]
6. Reduce to a simple integral the multiple integral
extended throughout the domain D defined by the inequalities
OS... o^x.. .... o<x., (5!y' + ... + (5.y"<i.
?•. DeriTe the formula
M
JJJ J r(|+i)
VII. Em.] exercises
where the multiple integral It eztMuitxi Uirougooui Um dooMln Ddttead ky Ite
Inequality
8*. Derive the formula
f (10 f F{a COB $ -k- b ain$ oo§^ -^ e tin $ Bin ^) tin 0d^stwf^^F{uil) dm,
where a, 6, and c are three arbitrary oonaUnu, and where B = VSHTPT?.
[PotaM*.]
[First observe that the given double integral la eqoal to a oartaia aorteee IbI»>
gral taken over the surface of the sphere 2* + y* + S* s 1. Thai take tba plaaa
tu; + ^2/ + cz = as the plane of zy in anew ayiteni of coOntlaaUa.]
9*. Let p = F(d, <f>) be the equation in polar oo<}nllnatee of a
Show that the volume of thn solid bounded by the eurfaoe ia equal to tte itrrahli
integral
{a) IjfpCOByda
extended over the whole surface, where da repreeenti the element of area, and 7
the angle which the radius vector makes with the exterior normaL
10*. Let us consider an ellipsoid whoae equation it
— 4- ' 4. = 1
and let us define the positions of any pohit on its sorfaoe by the aOipCle oo(MI-
nates v and p, that is, by the roots which the aboTO equation wonld lMf« K ^
were regarded as unknown (cf. § 147). The application of the liwnia (tt) ••
the volume of this ellipsoid leads to the equation
Likewise, the formula (a) gives
Jq Jb
V(6*-/^(ci-/^(»«-M)(«^-i^ • [La«i.)
11. Determine the funcUons P(x, y) and Q(x, y) which, togeClMr wlUi ihaU
partial derivatives, are continuous, and for which the line iniegrai
JPC* + a, y + « d« + Q(« + a, y + A) %
token along any closed contour whateTer ia tnJa p w i d wi t of Um ooMtanta a and
B and depends only upon the contour itself.
^ [Lk«c«, Pafta. July. 1900.]
826 MULTIPLE INTEGRALS [VII, Exs.
18^. Consider the point transformation defined by the equations
y = 0(x', y', 2') ,
s = f(x', 1/^,2').
As the point (a^, y*, zO describes a surface S% the point (x, j/, z) describes a sur-
face 5. Let a, ^, 7 be the direction angles of the normal to S ; a\ p\ Y the
direction angles of the corresponding normal to the surface 8' ; and da and da'
the corresponding surface elements of the two surfaces. Prove the formula
COS7d.=±d<r-{ j^^>^ cosa- + ^<^cos/3- + ^<^cosy}.
13*. Derive the formula (16) on page 304 directly.
[The volume V may be expressed by the surface integral
V= f z cosy da,
and we may then make use of the identity
D{x', y'^zf) dx^r D(y\ zf)) dir D{z\ X') ) dz'V B{x\ y') j
which is easily verified.]
CHAPTER VIII
INFINITE SERIES
I. SERIES OF REAL CONSTANT TERMS
GENERAL PROrERTIKS TESTS FOR CONVERGENCE
156. Definitions and general principles. Seqaenoet. The elementarj
properties of series aie discussed in all texU on College Algebra
and on Elementary Calculus. We shall review rapidly the principal
points of these elementary discussions.
First of all) let us consider an infinite tequence uf qumn titles
(1) «o, «i, «s
in which each quantity has a definit< rder of preoedeoea
hQm^ fixed. Such a sequence is said u, .. , -j^fU if #, approftehet
a limit as the index n becomes infinite. Every sequence which is
not convergent is said to be divergent. This may happen in either
of two ways : s^ may finally become and remain larger than any
preassigned quantity, or s^ may approach no limit even though it
does not become infinite.
In order that a sequence should be convergent, it is
sufficient thaty corresponding to any preoMigned poeUive nwmker c, a
positive integer n should exist such that the difference «,♦» — ^ *•
less than c in absolute value for any positive integer p.
In the first place, the condition is necessary. For if j^ approacfaat
a limit s as n becomes infinite, a number it always exists for whieh
each of the differences < — »,, * — *■>!» ••» *—**♦#> •••is lass than
</2 in absolute value. It follows that the absolute value of #, ^ ,
will be less than 2 c/2 = c for any value of p.
In order to prove the converse, we shall introdnoe a v«ry impor>
tant idea due to Cauchy. Suppose that the absolute value of eaeh
of the terms of the sequence (1) is less than a positive number AT.
Then all the numbers between — N and + N may be separated into
two classes as follows. We shall say that a number balangi lo Um
class .4 if there • vJ^» an infinite number of terms of the sequtnoe (1)
:W7
828 INFINITE SERIES [VIII, §166
which are greater than the given number. A number belongs to
the claai B if there are only a finite number of terms of the
sequence (1) which are greater than the given number. It is
evident that every number between - N amd -hN belongs to one
of the two classes, and that every number of the class A is less
than any number of the class B. Let S be the upper limit of the
numbers of the class A, which is obviously the same as the lower
limit of the numbers of the class B. Cauchy called this number the
greatest limit {la plus grande des limites) of the terms of the
sequence (1).* This number S should be carefully distinguished
from the upper limit of the terms of the sequence (1) (§ 68). For
instance, for the sequence
.111
^' 2' 3^ •"' n '"
the upper limit of the terms of the sequence is 1, while the greatest
limit is 0.
The name given by Cauchy is readily justified. There always
exist an infinite number of terms of the sequence (1) which lie
between S — € and .S -|- c, however small c be chosen. Let us then
consider a decreasing sequence of positive numbers ci, cg, •••,
c,, •••, where the general term «„ approaches zero. To each num-
ber €i of the sequence let us assign a number a,, of the sequence (1)
which lies between -S — €,• and S + Ci. We shall thus obtain a
suite of numbers ai, a^, •••, a„, ••• belonging to the sequence (1)
which approach S as their limit. On the other hand, it is clear
from the very definition of S that no partial sequence of the kind just
mentioned can be picked out of the sequence (1) which approaches
a limit greater than S. Whenever the sequence is convergent its
limit is evidently the number S itself.
Let us now suppose that the difference s„^p — s„ of two terms of
the sequence (1) can be made smaller than any positive number c
for any value of /> by a proper choice of n. Then all the terms of
the sequence past s^ lie between s„ — e and s„ -\- c. Let S be the
greatest limit of the terms of the sequence. By the reasoning just
given it is possible to pick a partial sequence out of the sequence (1)
which approaches S as its limit. Since each term of the partial
•aqaeoce, after a certain out, lies between s^ — c and s„ -\- «, it is
•n4mtm4» analyHques de TuHn, 19SS (CoUetted Works, 2d series, Vol. X, p. 49).
The daflnltloD may be extended to any aasemblage of numbers which has an upper
nsilt.
VI11,§1S7J CONSTANT TKKIIg S£9
clear that the absolute value of ^ — «. is at mott eqita] to c Nov
let «» be any term of the sequence (1) whoee index m is pnotm
than n. Then we may write
and the value of the right-hand side is surely less than 2c Sinea t
is an arbitrarily preassigned positive number, it follows that the
general term «. approaches 5 as its limit as the index m increases
indefinitely.
Note. If S is the greatest limit of the terms of the sequence (1),
every number greater than S belongs to the class B, and OTery num-
ber less than S belongs to the class A. The number S itnlf may
belong to either class.
157. Passage from sequences to series. Given any infinite saquenea
Wo, Wj, 1/„ .., tt,, -.-,
the series formed from the terms of this sequence,
(2) «o + «i + Wt + - • + «. + •••,
is said to be convergent if the sequence of the saocaaaiTa stuns
So = yo, Si = UQ + ni, ..., 5, = tto-f-tti + --- + ti„ •••
is convergent. Let S be the limit of the latter sequence, ie. the
limit which the sum S^ approaches as n increases indefinitely:
trmm
5 == lim 5, = lim V «».
"-• "-•^
Then S is called the sum of the preceding serieSf and this retatioii is
indicated by writing the symbolic equation
5 = Wo4-Wi +• -f w. -f •••=X"''*
A series which is not convergent is saia to oc diver^emL
It is evident that the problem of determining whalher the aariaa
is convergent or divergent is equivalent to the problem of determin-
ing whether the 8e<iuenoe of the successive sums ^, Si, ^, ••• ia
convergent or divergent. Conversely, the sequenoa
*0, 'l» *•> "'» *•'
will be convergent or divergent aoeording as the sariaa
5o -H (»i -«•) + («t - «i) +•+(«•-*—») + •••
380 INFINITE SERIES [VIII, §167
is convergent or divergent For the sum S^ of the first n + 1 terms
of this series is precisely equal to the general term s^ of the given
sequence. We shall apply this remark frequently.
The series (2) converges or diverges with the series
(3) w, + w,+i + --- + Wp+, + ---,
obtained by omitting the first p terms of (2). For, if S^(n >^)
denote the sum of the first » + 1 terms of the series (2), and 2„_p
the sum of the n — /> + 1 first terms of the series (3), i.e.
the difference 5, - 2,_^ = i^^ + wj + • • • + w^.i is independent of n.
Hence the sum \_p approaches a limit if 5„ approaches a limit,
and conversely. It follows that in determining whether the series
converges or diverges we may neglect as many of the terms at the
beginning of a series as we wish.
Let 5 be the sum of a convergent series, S^ the sum of the first
n + 1 terms, and R^ the sum of the series obtained by omitting the
first n -f 1 terms,
K = ^^ + 1 + Un + 2 H f ^n + p H •
It is evident that we shall always have
5 = 5„ + R,,
It is not possible, in general, to find the sum 5 of a convergent
series. If we take the sum S of the first n -{-1 terms as an approxi-
mate value of S, the error made is equal to R„. Since S„ approaches
5 as n becomes infinite, the error R^ approaches zero, and hence the
number of terms may always be taken so large — at least theoret-
ically — that the error made in replacing S by 5„ is less than any
preassigned number. In order to have an idea of the degree of
approximation obtained, it is sufficient to know an upper limit
of /?,. It is evident that the only series which lend themselves
readily to numerical calculation in practice are those for which
the remainder R^ approaches zero rather rapidly.
A number of properties result directly from the definition of con-
rergenoe. We shall content ourselves with stating a few of them.
1) y each of the terms of a given series be multiplied by a constant
k different from zerOf the new series obtained will converge or diverge
with the given series; if the given series converges to a sum S, the sum
qfthe seeond series is kS,
VIII. §138] CONSTANT TERlfg |g|
2) If there he given two eonvergent series
«o + Mi + «« + •• + «, + ...,
Vt4-Vi + », + - • + V. + --,
whose sums are S and S*, respeetivelf, the new smist eUaim^d hp
adding the given series term hy term,
K -f t;^) + (th + v,)+ ••• +(u. + tf.)+ •
converges, and its sum m 5 -f 5'. The analogous theorem hoUU for
the term-by -tenn addition of p convergent series.
3) The convergence or divergence of a series is not affeeted if the
values of a finite number of the terms he changed. For such a change
would merely increase or decrease all of the sums S^ after a oertain
one by a constant amount.
4) The test for convergence of any infinite sequence, applied to
series, gives Cauchy^s general test for convergence : •
In order that a series he convergent it is neeessarg and
thatf corresponding to any preassigned positive number c, a
n should exist, such that the sum of any number of terms what-
ever, starting with w,^.i, is less than c in abeoiute vaiu$. For
^n + p - Sn = ^^» + I + W,+, H + W«+p.
In particular, the general term w,^, = 5,^, — 5, must mppremek
zero as n becomes infinite.
Cauchy's test is absolutely general, but it is often difficult to
apply it in practice. It is essentially a development of the ftrj
notion of a- limit. We shall proceed to recall the practical rules most
frequently used for testing series for convergenoe and dirergeiiee.
None of these rules can be applied in all oases, bat together they
suffice for the treatment of the majority of cases which actually arise.
158. Series of positive terms. We shall oommenoe fay investigatiiif
a very important class of series. — those whose terms are all posi-
tive. In such a series the sum 5. increases with n, Henee in
order that the series converge it is sufficient thai the sum 5. should
remain less than some fixed number for all values of m. The most
general test for the convergence of such a series is based upon com-
parisons of the given series with others prerionsly studied. Tb*
following propositions are fundamental for this process:
^£x«rctoM de iral/UmoMYMif, 1817. (OoUmCmI ITtrte, VoL Til, M
8S2 INFINITE SERIES [vm,§159
1) If each of the terms of a given aeries of positive terms is less
than or at most equal to the corresponding term of a known convergent
weriet of positive termsy the given series is convergent For the sum
5, of the first n terms of the given series is evidently less than the
sum S* of the second series. Hence S^ approaches a limit S which
is less than S\
2) If each of the terms of a given series of positive terms is greater
than or equal to the corresponding term of a known divergent series
of positive termSy the given series diverges. For the sum of the first
n terms of the given s^ies is not less than the sum of the first
n terms of the second series, and hence it increases indefinitely
with n.
We may compare two series also by means of the following
lemma. Let
(U) Mo + Wj + w, + ... + w„4....,
(F) t;^-|-Vj+t;^ + ...+V^ _!_...
be two series of positive terms. If the series (U) converges, and if,
after a certain term, we always have v„+i/v„ ^ Wn+i/w„, the series ( V)
also converges. If the series (U) diverges, and if, after a c&rtain
term, we always have u^+i/u„<v^+i/v„, the series (F) also diverges.
In order to prove the first statement, let us suppose that
^•-^i/^'K+i/^n whenever n>p. Since the convergence of a
series is not affected by multiplying each term by the same con-
stant, and since the ratio of two consecutive terms also remains
unchanged, we may suppose that v^ < u^, and it is evident that we
should have v^^,<u^^^, Vp^,<u^^„ etc. Hence the series (F)
must converge. The proof of the second statement is similar.
Given a series of positive terms which is known to converge or
to diverge, we may make use of either set of propositions in order
to determine in a given case whether a second series of positive
terms converges or diverges. For we may compare the terms of
the two series themselves, or we may compare the ratios of two
eonsecutiye terms.
159. Ctuchy's test and d'Alembert'B test. The simplest series which
OMi be used for purposes of comparison is a geometrical progression
whose ratio is r. It converges if r < 1, and diverges if r > 1. The
eomptrison of a given series of positive terms with a geometrical
progression leads to the following test, which is due to Cauchy:
VIII, 5159] CON'STAN'T TERMR $|g
If the nth root "wu^ qf the general term u^ qf a smrise ^
terms after a certain term is eonatanily U$s tJkmi a fiaitd
than unity, the series converges. If y/u^ after a Mvioiw term is cmi-
stantly greater than unity, the series diverges.
For in the first case 'Vu,<k<l, whence M.<Jb". Heooe eeeh
of the terms of the series after a certain one ie leti than the oon^
sponding term of a certain geometrical progreeeion whoee ratio ie
less than unity. In the second case, on the other hand, \^>1,
whence u^>l. Hence in this caae the general term doee not
approach zero.
This test is applicable whenerer "V^ i^proachea a limit Id
fact, the following proposition may be stated:
If v^ approaches a limit I as n heeomes infinite, ths ssriss wiU
converge if I is less than unity j and it will diverge \flis greater tAsm
unity.
A doubt remains if lalj except when "Vu^ remains greaUr ikms
unity as it approaches unity, in which mm the series surely div erfs t.
Comparing the ratio of two consecatiye terms of a given series
of positive terms with the ratio of two consecutiTe termi of a
geometrical progression, we obtain d'Alembert's test:
If in a given series of positive terms the ratio of any term te tMe
preceding after a certain term remains less than a fixed
less than unity, the series converges. If thai raiie softer •
term remains greater than unity, the series div erg es *
From this theorem we may deduce the following corollary:
If the ratio u^^x/u^ approaches a limit I as n heoomee infinite, the
series converges if I <!, and diverges tfl>l.
The only doubtful case is that in which / « 1 ; a»eii then, i^t^4.i/*^
remains greater than unity as it approaehes umiig, the series is diesrge^
General commentary. Cauchy*s fast is BOffe gMMnl tbaa d*AlMBbert*t. Wee
suppose that the terms of a given miM, iflar a osftain one, art seek Its Ikea
the corresponding terms of a decreasliig gsooMliieal iNUgiMrinn. Le. that Ike
general term u« is leas than Ar* for all TihiM of a grealertheaa ftsed liilli»F>
where A is a certain constant and r is lea than nnhy. Henoe V^ < MW*, aad
the second member of thia inequality a|>proeohM uatty •• m bMomta leMeSm.
Hence, denoting by fc a fixed number between r and 1, we shall have after a eer-
tain term Vu^<k. Hence Cauchy's test Ie appUeable In any aneh eeee. Bal ll
may happen that the raUo u, + i/m, assumes ralnee fraater than unity.
far out in the series we may go. For example, consider the
l + r|alna| + r«|ain2a| + .-- + »«|sln«a| +
3g4 INFINITE SERIES [VIII, §169
where r < 1 and where a ia an arbitrary conatant. In this case V^ = r V|sinna| < r,
whereat the ratio
Un±i_^ 8in(n + l) a
tin sin na
may aKome, in general, an infinite number of values greater than unity as n
iocreaeee indefinitely.
Nererthelen, it is advantageous to retain d'Alembert's test, for it is more
convenient in many cases. For instance, for the series
1 1.2 1.2.3 1.2-. n
the wUo of any term to the preceding is x/{n +1), which approaches zero as n
becHwneff infinite ; whereas some consideration is necessary to determine inde-
pendently what happens to Vun = x/\/l . 2 • • • n as n becomes infinite.
After we have shown by the application of one of the preceding tests that each
of the terms of a given series is less than the corresponding term of a decreasing
geometrical progression A, Ar^ Ar^, • • • , ^r», • • • , it is easy to find an upper
limit of the error made when the sum of the first m terms is taken in place of
the sum of the series. For this error is certainly less than the sum of the
geometrical progression
Ar^
Ar^ + Ar^+^ + A7^+^ + '-' =
1 — r
When each of the two expressions -n/w^ and Vn + i/Un approaches a limit, the
two limits are necessarily the same. For, let us consider the auxiliary series
(4) Uo-\-UiX-\- UiZ^ + • • • + lAnX** + • • • ,
where x is positive. In this series the ratio of any term to the preceding
approaches the limit Ix, where I is the limit of the ratio w„ + i/m„. Hence the
■eriee (4) converges when x < 1/i, and diverges when x > l/l. Denoting the
limit of v^ by T, the expression y/unX'* also approaches a limit Vx, and
the series (4) converges if x < l/l\ and diverges if x > 1/V. In order that the
two tests should not give contradictory results, it is evidently necessary that I
and /' should be equal. If, for instance, I were greater than T, the series (4) would
be convergent, by Cauchy's test, for any number x between l/l and 1/V, whereas
the aame series, for the same value of x, would be divergent by d'Alembert's test.
Still more generally, if tt, + i/u„ approaches a limit Z, v^ approaches the same
limit.* For suppose that, after a certain term, each of the ratios
^ + 1 Un + t _ Un + p
llee between I - c and I + e, where e is a positive number which may be taken
M tmaU as we pleaee by uking n sufficiently large. Then we shall have
u:*''(i-.)-+p< 'yy^^^ < u:+p(i + .)-+i>.
•Cauchy, Court d* Analyse,
VIII, f 160] CONSTANT TSRMS t|6
As the number p inoretaM indefinitely, while » remaiiMi ftsed, the two leran oa
the extreme right and left of Uiie double inequaUty appriMeh I -f • Mid I ~ c,
respectively. Hence for all raluee of m greater than a nilablj
we ihall have
and, since e is an arbitrarily assigned number, it follows that K(m^
tlie number / as it« limit.
It should be noted that the converse is not true. Oonalder, for
sequence
1, a, 06, cflby a«6«, •., o"ft^-», ••6>», ,
where a and b are two different numbers. The ratio of any term to the praeed-
ing is alternately a and 6, whereas the expression \(u^ approaehss the Uailt V3
as n becomes infinite.
The preceding proposition may be employed to determine the UmUs of Mr-
tain expressions which oc cur in undetermined forms. Thos it Is srldeBt that
the expression v^l .i-n increases indefinitely with a, since the ratio a !/(a - 1)1
increases indefinitely with n. In a similar manner it may be shown that each of
the expressions -v^ and \^logn approachee the limit unity as a
160. Application of the greatest limit. Cauchy formolated the
in a more general manner. Let eu be the general term of a
terms. Consider the sequence
1 1 !
(5) au <4, fl{, •••, K* ••••
If the terms of this sequence have no upper limit, the gSMral
approach zero, and the given series will be divergenL If aU the tenas of the
sequence (5) are less than a fixed number, let w be the gi es t es l limit of the larms
of the sequence.
The series Za^ is convergent iftiUUta ikom wiif|f, ami M wmrgm t ifttiM
than unity.
In order to prove the first part of the theorem, let 1 - a be a aomber
w and 1. Then, by the definition of the greatest limit, there exist bat a
number of terms of the sequence (6) which are greater than 1 - a. It follows
that a positive integer p may be found such that •Voi < 1 - a for all values of a
greater than p. Hence the series So. converges. On the other hand. If •» > I,
let 1 + a be a number between 1 and «. Then there are an Infiaita aambsr ef
terms of the sequence (6) which are greater than I + a, and heooe there are aa
infinite number of values of n for which o. is greater than nnl^. U foOofis that
the series Sa, is divergent in this case. Tbeeass in which •• « 1 rsmatos ladowht.
161. Cauchy 's theorem. lu case Hi^i/i*. wd Vu^ both approMh
unity without remaining constantly gnater than unity, neither
d'Alembert's test nor Cauchy's test enables tie to decide whether
the series is convergent or divergent We moft then take M a
comparison series some series which has the sane oharaMerMo
886 INFINITE SERIES [VIII, § 161
but which is known to be convergent or divergent. The following
proposition, which Cauchy discovered in studying definite integrals,
often enables us to decide whether a given series is convergent or
divergent when the preceding rules fail.
Let ^(j-) be a function which is positive for values of x greater
than a certain number a, and which constantly decreases as x
increases past x — a, approaching zero as x increases indefinitely.
Then the x axis is an asymptote to the curve y = </>(x), and the
definite integral
jT
<f>(x)dx
may or may not approach a fiinite limit as I increases indefinitely.
The series
(6) i>(a) 4- ,^(a + 1) + • • • + <^(a + w) + • . .
converges if the preceding integral approaches a limit, and diverges if
it does not.
For, let us consider the set of rectangles whose bases are each
unity and whose altitudes are <f>{a), <f>{a +1), • • ., <^(a + n), respec-
tively. Since each of these rectangles extends beyond the curve
y = ^(x), the sum of their areas is evidently greater than the area
between the x axis, the curve y = <i>{x), and the two ordinates x = a,
jc = a -f n, that is,
<t>(x)dx.
On the other hand, if we consider the rectangles constructed
inside the curve, with a common base equal to unity and with the
altitudes «^(a 4-1), <t>{a -f 2), • • ., <^(a + n), respectively, the sum of
the areas of these rectangles is evidently less than the area under
the curve, and we may write
^(a) 4- ^(a +1) -f • • • + ^(a 4- n) < <^(a) 4- f" ^<f>(x)dx.
Hence, if the integral X'<^(a;) rfa; approaches a limit L as I increases
indefinitely, the sum ^(a) + ... +^(a-{-n) always remains less than
^(a) 4- L, It follows that the sum in question approaches a limit ;
hence t^ series (6) is convergent. On the other hand, if the inte-
ff^f^^*H^)dx increases beyond all limit as n increases indefinitely,
the same is true of the sum
*(«) 4- ^(a 4- 1) 4- • • • -f <^(a 4- n),
VIII, § 161] CONSTANT TERMS 817
as is seen from the first of the above inequAlitiat. Hene^ in thif
case the series (G) diverges.
Let us consider, for example, the function 4(x)v]/j^, whn% ft
is positive and a =1. This function satisfies all the requiienms
of the theorem, and the integral j^\\/af^'\dx appxoacbee a limit ••
/ increases indefinitely if and only if ft is greater than unity. It
follows that the series
p + 2^ + 3;r + --- + jjj + ---
converges if /i is greater than unity, and diverges if ft <1.
Again, consider the function ^(2)sl/[x(logx)''], where log*
denotes the natural logarithm, /a is a positive number, and awm%.
Then, if /* =^ 1, we shall have
X'
dx —1
___ = __[(log„).-._(,og2).-].
The second member approaches a limit if ft > 1^
indefinitely with n if /a < 1. In the particular case when /kmI it
is easy to show in a similar manner that the integral
beyond all limit. Hence the series
2(log 2Y ^ 3(log 3r ^ ^ n(log n^ ^
converges if /i > 1, and diverges if fi<l.
More generally the series whose general term is
1
n log n log« n log* n • • 1(^'~* »(log' nf
converges if /i > 1, and diverges if /* ^ 1. In this expression log*»
denotes log log n, log' rt denotes log log log n, etc. It is understood,
of course, that the integer n is given only valoes so large that
logn, log^w, log«n, ••, log^n are positive. The missing terat in
the series considered are then to be supplied by aeros. The
theorem may be proved easily in a manner similar to the demoo-
strations given above. If, for instance, fi¥»l, the funotioo
1
X log a; log* X. •(log' 'V*
is the derivative of (log" x)> -**/(! - m)» ^^ ^^ *»«^^ tuncttoo
approaches a finite limit if and only if ^ > 1.
888 INFINITE SERIES [VIII, §162
Caocby*! theorem admits of applications of another sort. Let us suppose
that the function ^(z) satisfies the conditions imposed above, and let us con-
^(n) + 0(n + 1) + . . . + ^(n + p),
where ii and p are two integers which are to be allowed to become infinite. If the
whoM general term is 0(n) is convergent, the preceding sum approaches
•• a limit, since it is the difference between the two sums Sn + p+i and Sn,
of which i^proaches the sum of the series. But if this series is divergent,
no conclusion can be drawn. Returning to the geometrical interpretation given
ahove, we find the double inequality
JT" %{x)dx < 0(n) + 0(n + 1) 4- • • • + 0(n + p) < <f>{ny+ f'"^%{z)dx.
Since 0(n) approaches zero as n becomes infinite, it is evident that the limit of
the sum In question is the same as that of the definite integral f*^'^^<p{x)dx,
and this depends upon the manner in which n and p become infinite.
For example, the limit of the sum
\ 1 +...+ »
n n + 1 n + p
to the nme ae that of the definite integral f^'^^ [1/x] dx = log(l + p/n). It is
etear that this integral approaches a limit if and only if the ratio p/n approaches a
limit. If a is the limit of this ratio, the preceding sum approaches log (1 + a)
M Ite limit, as we have already seen in § 49.
Finally, the limit of the sum
Vn Vn + l Vn+p
ft the aame as that of the definite integral
s:
la order that this expression should approach a limit, it is necessary that the
ratio p/Vn should approach a limit or. Then the preceding expression may be
written hi the form
= 2
P
yTn
and It la erident that the limit of this expression is a.
192, Logarithmic criteria. Taking the series
M a omnparison series, Caucliy deduced a new test, for convergence
wWoh U entirely analogous to that which involves </u^.
VIII, §162] CONSTANT TKRMS ggg
If after a certain term tht- tj-jir-' \ ',)/logJi if alwayg
greater than a fixed number which t^ j,''". r i',<in unity ^ ike serts»
converges. If after a certain term log(l/Mj/logn w aUoojfg Ut$
than unity y the series diverges.
If log(l/tt^)/logn approa^shes a limit t <u n increases ind^nMf,
the series converges if / > 1, and diverges if / < 1, The ease m
which / = 1 remains in doubt.
In order to prove the first part of the theorem, we will remaik
that the inequality
log— > k log n
is equivalent to the inequality
— > n* or w_ < -r ;
"since A; > 1, the series surely converges.
Likewise, if
log— < log n,
we shall have ?/„ > I/71, whence the series surely diverges.
This test enables us to determine whether a given series con-
verges or diverges whenever the terms of the series, after a certain
one, are each less, respectively, than the corresponding terms of
the series
wh
ere A is a
constant factor and /a > 1. For,
if
''-^^'
we
shall have
5 log u^
4- /x log n < log A or
logn ^ ^ logn'
and the right-hand side approaches the limit /ui as n im
indefinitely. If A' denotes a number between unity and ^ we
shall have, after a certain term,
rr—^>K.
logn
340 INFINITE SERIES [VIII,§163
Similarly, taking the series
Z^ n(log nY ' Z^ n log w(log'* w)" '
as comparison series, we obtain an infinite suite of tests for con-
vergence which may be obtained mechanically from the preceding
by replacing the expression log(l/w„)/log7i by log [l/(7iw„)]/ log^ ;i,
then by
, 1
log \
nu^ log n
i
log'w
and so forth, in the statement of the preceding tests.* These tests
apply in more and more general cases. Indeed, it is easy to show
that if the convergence or divergence of a series can be established
by means of any one of them, the same will be true of any of those
which follow. It may happen that no matter how far we proceed
with these trial tests, no one of them will enable us to determine
whether the series converges or diverges. Du Bois-Reymond f and
Pringsheim X have in fact actually given examples of both convergent
and divergent series for which none of these logarithmic tests deter-
mines whether the series converge or diverge. This result is of great
theoretical importance, but convergent series of this type evidently
converge very slowly, and it scarcely appears possible that they
should ever have any practical application whatever in problems
which involve numerical calculation. §
163. Raabe's or Duhamel's test. Retaining the same comparison
series, but comparing the ratios of two consecutive terms instead
of comparing the terms themselves, we are led to new tests which
are, to be sure, less general than the preceding, but which are
often easier to apply in practice. For example, consider the series
of positive terms
(7) Wo + «^ + wj H h w« H ,
• B«6 Bertnuid, Traiti de Calctd diffirerUiel et integral, Vol. I, p. 238; Journal
dt lAouviUe, Ut series, Vol. VII, p. 35.
t Ueber Cotwergmx von ReUien . . . (Crelle'a Journal, Vol. LXXVI, p. 85, 1873).
I Attgnnelnt Tfuorie der Divergenz . . . (Sfathematische Annalen, Vol. XXXV,
1800).
f In an example of a certain convergent series due to du Bois-Reymond it would
bs nuc essa r y, according to the author, to take a number of terms equal to the volume
<^f Ihe §arth eiqnreMted in cubic millimetert in order to obtain merely half the sum of
Vlli,§ia3j CONSTANT TERMS S41
in which the ratio u.^i/u, approaches unity, remaining coiuuuiuj
less than unity. Then we may write
where a. approaches zero as n becomes infinite. The oomparisoD of
this ratio with [n/{n -hi)]** leads to the following rule, discovered
first by Raabe* and then by Duhamel.f
If after a certain term the product na^ u alwayt grtaUr iktm m
fixed number which is greaier than unity, the serisi eamver^. If
after a certain term the same product is always Uu than uniiyf tk$
series diverges.
The second part of the theorem follows immediately. For, since
na^ < 1 after a certain term, it follows that
1 n
and the ratio u^ + i/u^ is greater than the ratio of two conteeutiTe
terms of the harmonic series. Hence the series diverges.
In order to prove the Urst part, let us suppose that after a certain
term we always have na^> k>l. Let /x be a number which lies
between 1 and k, l</l<^^ Then the series surely converges if
after a certain term the ratio u.^i/u. is less than the ratio
[n/(n -1-1)]'' of two consecutive terms of the series whoee general
term is n'*^. The necessary condition that this should be true
is that
(8) • ' - '
l+«.
M'
or, developing (1 -f l/n)** by Taylor's theorem limited to the term
in l/7tS
where A^ always remains less than a fixed number as n becomes
infinite. Simplifying this inequality, we may write it in the fom
• ZsUsohri/i /Br MaihmnatUt uttd Pk^ttk, Vol. Z,
i Journal ds LknwOls, Vol. IV, 1888.
842 INFINITE SERIES [VIU, §163
The left-hand side of this inequality approaches /a as its limit as n
becomes infinite. Hence, after a sufficiently large value of n, the
left-hand side will be less than na^^ which proves the inequality (8).
It follows that the series is convergent.
If the product na^ approaches a limit ^ as w becomes infinite, we
may apply the preceding rule. The series is convergent if Z>1,
and divergent if /<1. A doubt exists if 1=1, except when na^
approaches unity remaining constantly less than unity : in that case
the series diverges.
If the product nan approaches unity as its limit, we may compare the ratio
u«4.i/u» with the ratio of two consecutive terms of the series
1.1.
2 (log 2)'* n(logw)'^
which converges if /i>l, and diverges if ti<\. The ratio of two consecutive
terms of the given series may be written in the form
i*»i + . 1
n n
where /3„ approaches zero as n becomes infinite. If after a certain term the
product /3« log n is always greater than a fixed number which is greater than unity y
the aeries converges. If after a certain term the same product is always less than
unity, the aeries diverges.
In order to prove the first part of the theorem, let us suppose that /3„ log n > A: > 1.
Let M be a number between 1 and k. Then the series will surely converge if after
a certain term we have
w„ n + 1 Llog(n + l)J *
which may be written in the form
n n \ n/ L logn J
or, applying Taylor's theorem to the right-hand side,
n n \ n/( logn "L logn J )
wiMre Xa always remains less than a fixed number as n becomes infinite.
Simplifying this inequality, it becomes
/ i\ ^«(n4-l)ri<>gCl + i')f
Ailogm>M(n + l)log(l+M+-l- iLlLIiiLL.
\ n/ logn
VIII, JlG.i] CONSTANT TERMS M^
The product (n + 1) log(l -f i/n) approaches uitiiy m n btooow iBflallA, lor U
may be writt«u, by Taylor*! theorem, in the form
(10)
(n + l)log/l + i^ = 1 + 1(1 + ,).
where « approaches zero. The right-hand aide of the above meqaalitj limnian
approaches fiMlia limit, and the truth of the inequality is nlililrtnil for nM
cieiitly large values of n, since the left-hand tide ia greater than It, wbieb it HhU
greater than fi.
The second part of the theorem may be proved by oompartng the rafk>
Un + i/Un with the ratio of two consecutive terma of the aerlea whoM fMietal
term is l/(nlogn). For the inequality
u, n + 1 log(n + l)'
which is to be proved, may be written in the form
n \ n/L logn J
n n \ n/L log i
/S„logn<(n + l)log
(-!)■
The right-hand side approaches unity through valuea which are greater than
unity, as is seen from the equation (10). The truth of the ineqoali^ is tbera*
fore established for sufficiently large valuea of n, for the Muhand ikla eaBBOl
exceed unity.
From the above proposition it may be shown, as a ooroUaiy, that If the piod-
uct /3n log n approaches a limit I as n becomes infinite, the l e ri ea ouu f ei gea if I >1,
and diverges if ^ < 1. The case in which / = 1 remains in doabt, unlea ^ loga
is always less than unity. In that case the series sorely divergea.
If /3nlogn approaches unity through values which are greater than untey, wa
may write, in like manner,
Uw-t-l __ 1
«. ,+ ' + ' + ^-*
n A logn
where y^ approaches zero as n becomes infinite. It would then be posribia to
prove theorems exactly analogous to the above bj o on a hfcifi ng the pcodMI
7„ log* n, and so forth.
,. Corollary. If in a series of posiUve terms the ratio of any term to the pra-
c^ej^ng can be written in the form
where m is a positive number, r a constant, and J?, a quantity whoae aha
value remains less than a fixed number as n inereaaaa IndeOiihaly, (At tmlm
verges if r ia greaUr than unity, and din^rgm te all Uker
844
INFINITE SERIES [vm,§164
For If we irt
u«+i_ 1
we ihall haye
Hn
u* + i
1
Wn
1+i
n
logn
n
n
n+1
n
^" n^
,logn =
1-
n
»! + **
and hence llm lur, = r. It follows that the series converges if r > 1, and diverges
If r < 1. The only case which remains in doubt is that in which r = 1. In order
to decide this case, let us set
From this we find
and the right-hand side approaches zero as n becomes infinite, no matter how
small the number fx may be. Hence the series diverges.
Suppose, for example, that Un + i/Un is a rational function of n which ap-
proaches unity as n increases indefinitely:
Un-n _ nP 4- aiTU^^ -f (H^P"^ H
Un ~ TIP + binP-^+b»7H^'^-\ *
Then, performing the division indicated and stopping with thfrtatm in 1/n*, we
may write
u„ + i 1 . ai-h <f>{n)
= 1 H 1 — ,
Un n n^
where 4>{n) is a rational function of n which approaches a limit as n becomes
infinite. By the preceding theorem, the necessary and sufficient condition that
the teriea should converge is that
6i > ai + 1 .
This theorem is due to Gauss, who proved it directly.* It was one of the first
for convergence.
164. Abtolute convergence. We shall now proceed to atudy series
whose tenns may be either positive or negative. If after a certain
term all the terms have the same sign, the discussion reduoes to
the previous caae. Hence we may restrict ourselves to series
which contain an infinite number of positive terms and an infinite
* (OotUeUd Works, Vol. Ill, p. 138.) Disquisitioneit generates circa seriern ii\finitam
a. a
Mil, S m] CONSTANT TERMS S46
number of negative terms. We shall prore first of all the fol-
lowing fundamental theorem :
Ani/ series whatever is eanvergeni if the §&ri§i farmed ^f the akee-
lute values of the terms of the given series eetwmr^ee.
Let
(11) «o-HMi-h---f K,4-
be a series of positive and negatiye terms, and lei
(12) f/^+f/j + .-.+ r, :
be the series of the absolute values of the terms of the given seneg,
where U^ = | Wh |- If the series (12) converges, the series (11) like-
wise converges. This is a consequence of the general theorem of
§ 157. For we have
and the right-hand side may be made less Uuui any prea.s8igned num-
ber by choosing n sufficiently large, for any subeequant choice of p.
Hence the same is true for the left-hand side, and the aeries (11)
surely converges.
The theorem may also be proved as follows : Let us write
and then consider the auxiliary series whose general term is n, -f- T,,
(13) (u, 4- ^o) + (1^ H- f^i) + ••• + (h, + tr.)+ •••.
Let S^y S'^j and S'^ denote the sums of the first n terms of the series
(11), (12), and (13), respectively. Then we shall have
The series (12) converges by hypothesis. Hence the series (IS)
also converges, since none of its terms is negative and its general
term cannot exceed 2U^. It follows that each of the sums ^^ and
S'^, and hence also the sum 5., approaches a limit as % increases
indefinitely. Hence the given series (11) converges. It is evident
that the given series may be thought of as arising from the guhtrsA-
tion of two convergent series of positive terms.
Any series is said to be absolutely o o nv e rgmi if the series of the
absolute values of its terms converges. In smek a series the order ^
the terms may be changed in any way whatever %eithoui eJUrimg the
846 INFINITE SERIES [VIII, §164
sum of the series. Let us first consider a convergent series of posi-
tiye termSi
(14) ao + «iH -l-a«H 7
whose sum is Sy and let
(16) *o + ^+-+*n4---
be a series whose terms are the same as those of the first series
arranged in a different order, i.e. each term of the series (14) is to
be found somewhere in the series (15), and each term of the series
(15) occurs in the series (14).
Let S'^ be the sum of the first m terms of the series (16). Since
all these terms occur in the series (14), it is evident that n may be
chosen so large that the first m terms of the series (15) are to be
found among the first n terms of the series (14). Hence we shall have
which shows that the series (15) converges and that its sum S' does
not exceed S. In a similar manner it is clear that S ^ S'. Hence
S' = S. The same argument shows that if one of the above series
(14) and (15) diverges, the other does also.
The terms of a convergent series of positive terms may also be
grouped together in any manner, that is^ we may form a series each
of whose terms is equal to the sum of a certain number of terms of
the given series without altering the sum of the series* Let us first
suppose that consecutive terms are grouped together, and let
(16) ^, + ^^+^, + ...4.^^ + ...
be the new series obtained, where, for example,
^0 = ^0 + «! + ••• + s, ^1 = S + i + • • • + a,,
Then the sum S'^ of the first m terms of the series (16) is equal to
the sum Sy of the first N terms of the given series, where N > m.
As m becomes infinite, N also becomes infinite, and hence S'^ also
approaches the limit S.
Combining the two preceding operations, it becomes clear that any
conmergerU series of positive terms muy be replaced by another series
each of whose terms is the sum of a certain number of terms of the
given series taken in any order whatever, without altering the sum of
•UiM ofton Mid that parentheaes tnay be inserted in a convergent series 0/ positive
in any inamur whatwer without altenng the sum 0/ the «erie«. — Trans.
VIII. §lfi.n] CONSTANT TFRMS 34J
the series. It is only ueoessary that each term of the giTen mtIm
should occur in one and in only one of the vroupM whii h form the
terms of the second series.
Any absolutely convergent series may be regarded as the differ-
ence of two convergent series of positive terms ; henee the praoedtiif
operations are permissible in any such series. It is evident that tn
absolutely convergent series may be treated from the point uf view
of numerical calculation as if it were a sum of a finite number of
terms.
165. Conditionally convergent series. A series whose terms do not all
have the same sign may be convergent without being absolutely eon-
vergent. This fact is brought out clearly by the following theoiem
on alternating series, which we shall merely state, ftffuming that it
is already familiar to the student.*
A series whose terms are alternately positive and negative convergee
if the absolute value of each term is less than that of the preeedimgp
and if in addition^ the absolute value of the terms of the series
diminishes indefinitely as the number of terms increases indefinitely.
For example, the series
converges. We saw in § 49 that its sum is log 2. The series
of the absolute values of the terms of this series is precisely the
harmonic series, which diverges. A series which oociTerges but
which does not converge absolutely is called a eonditienaUy eemeet'
(jent series. The investigations of Cauchy, Lejeune-Dirichlet, and
Kiemann have shown clearly the necessity of distinguishing between
absolutely convergent series and conditionally convergent aeriet.
For instance, in a conditionally convergent series it is no^ always
allowable to change the order of the terms nor to group the terms
together in parentheses in an arbitrary manner. These operatioos
may alter the sum of such a series, or may change a ooDraifaat
series into a divergent series, or vice versa For exampla. l«t ui
again consider the convergent series
111 1 1
2^3 4^ ^2n+l 2ii + 3
• It is iM>int^ out in § 166 that this therein to % wgeOsl esse of Um tlMoraai pnm4
there. — Trans.
848 INFINITE SERIES [vm,§l66
whose Bum ia evidently equal to the limit of the expression
as m becomes infinite. Let us write the terms of this series in another
Older, putting two negative terms after each positive term, as follows :
^ 2 4^3 6 8^ ^2n+l 4n + 2 4w + 4^
It is easy to show from a consideration of the sums S^^, S^^^^, and
•^ta^i *^** *^® ^®^ series converges. Its sum is the limit of the
expression
'vi^ I 1-]
.T!)\2^+1 4n + 2 4w + 4/
as m becomes infinite. From the identity
_JL 1 1_ = 1 /_J_ _ 1 \
2n4-l 4n-f2 4w + 4 2\27i-hl 2n-\-2/
it is evident that the sum of the second series is half the sum of
the given series.
In general, given a series which is convergent but not absolutely convergent,
It is possible to arrange the terms in such a way that the new series converges
toward any preassigned number A whatever. Let Sp denote the sum of the
first p positive terms of the series, and Sg the sum of the absolute values of the
first q negative terms, taken in such a way that the p positive terms and the q
negative terms constitute the first p + q terms of the series. Then the sum of
the first p + q terms is evidently Sp — Sg. As the two numbers p and q increase
indefinitely, each of the sums Sp and S'g must increase indefinitely, for otherwise
the series would diverge, or else converge absolutely. On the other hand, since
the series is supposed to converge, the general term must approach zero.
We may now form a new series whose sum is A in the following manner :
Lei OS take positive terms from the given series in the order in which they occur
in it until their sum exceeds A. Let us then add to these, in the order in which
they occur in the given series, negative terms until the total sum is less than A.
Again, beginning with the positive terms where we left off, let us add positive
trnms until the total sum is greater than A. We should then return to the
Mgative terms, and so on. It is clear that the sum of the first n terms of the
new series thus obtained is alternately greater than and then less than A, and
that it differs from ^ by a quantity which approaches zero as its limit.
IM. Absl's test The following test, due to Abel, enables us to establish the
eoBTergenoe of certain series for which the preceding teste fail. The proof is
' apon the lemma sUted and proved in § 76.
JM
«o + wi -H . . . -t u„ -f • • •
VIII, $166] CONSTANT TERMS S49
be a series which conTorgat or which if imMa wfaBlt (that !■• for wUdi Qm na
of tiie first n t4;rinH in always leas than a flzed nombtr it In *»ir*^it ▼■!■•).
Again, let
Co, «i, •••, c., • •
be a monotoDically decreasing seqnenoe of poaitiTe nnmbers which apprmeb
zero as n becomes infinite. Then the
(17) «oMo + *iKi+ •• + <.!«• + •••
converges under the hypothean made abofte.
For by the hypotheses made above it foDows that
for any value of n and p. Hence, by the lemma jntt referred to, wo may writ*
Sinc^ cm+i approaches zero as n becomes infinite, n may be chosen so large that
the absolute value of the sum
<«+iWi. + i 4- • • • + €,+,u,^.^
will be less than any preassigned positive number for all values of p. Tho
series (17) therefore converges by the general thecnvm of | 167.
When the series Uo+UiH l-u, + --- reduces to the series
1_ 1 + 1-1 4-1 _ 1
whose terms are alternately +1 and — 1, the thf >rfm -•( tln^ .»:ir i»- r. -....». u>
the theorem stated in § 165 with regard to alternating aeries.
As an example under the general theorem consider the series
8intf + sin2tf + 8in8* + .- + sinn#-|--.
which is convergent or indeterminate. For if sin # s 0, every term of tba aariM
is zero, while if sin 6^0, the sum of the first n terms, by a formaU of Trigo-
nometry, is equal to the expression
sin —
^..(--±1,),
2
which is less than | l/sin {6/2) \ in absolute value. It follows that tba aariaa
sin* rin2f ^ •toi*# ^ ..
1 2 »
converges for all values of $. It may be shown In a similar manner that the
series
ooetf ooe2f . ooan#
+ — — — + • • • + _ ▼ • • •
2 n
converges for .ill v.imts vi 4 except $ = 2kw»
860 INFINITE SERIES [VIII, §167
Corollary. Restricting ourselves to convergent series, we may state a more
general theorem. Let
be a conrergent series, and let
be any monotonically increasing or decreasing sequence of positive numbers
which approach a limit k different from zero as n increases indefinitely. Then
the ieriea
(18) «0W0 + «lWl H + enlLn + '"
aUo converges.
For definiteness let us suppose that the c's always increase. Then we may
write
eo = k-ao, €i = k — ai, • • •, €„ = A; - a„, ••-,
where the numbers ao , o'l , • • • , <3'n , • • • form a sequence of decreasing positive
numbers which approach zero as n becomes infinite. It follows that the two
series
kuo-h kui + hkiLn + • • • ,
aoUo + rt'iMi H + a„Un H
both converge, and therefore the series (18) also converges.
II. SERIES OF COMPLEX TERMS MULTIPLE SERIES
167. Definitions. In this section we shall deal with certain gen-
eralizations of the idea of an infinite series.
Let
(19) Uq -{- Ui -\- ut -{ \-u„-\
be a series whose terms are imaginary quantities :
«o = ao + *o**> Ui = ai + bii, •••, w„ = a„ + ^„i,
Such a series is said to be convergent if the two series formed of
the real parts of the successive terms and of the coefficients of the
imaginary parts, respectively, both converge:
(20) tto + «! + a« + • • • + a„ + • • • = 5', .
(21) 6o+*i+ft, H----+^»„ + --- = 5".
Let 5' and S" be the sums of the series (20) and (21), respectively.
Then the quantity 5 = 5' -f iS" is called the sum of the series (19).
It is evident that S is, as before, the limit of the sum S^ of the first
n terms of the given series as n becomes infinite. It is evident
that a series of complex terms is essentially only a combination of
two series of real terms.
VIII, §168] COMPLEX TERMS MULTIPLE ftgttrM $5^
When the series of ahtoluU values of the terms of the serim (19)
(22) V«! + « + >/«} + ^ + - •+v/5T^.4-.
convergesy each of the ser ies (20) a^id (21) evid emilff
lutely, for \a,\ S \/«i + *! and \K\ i N/ajV ^.
In this case the series (19) is said to be absoltUehj etmverg eni . The
sum of such a series is not altered by a change in the order ef the
terms f nor by grouping the terms together in any way.
Conversely t if each of the series (20) an d (21) e o nv erge e aheoimtely,
the series (22) converges absolutely f for v«J + ^ 5 l^*.! + |^«|»
Corresponding to every test for the convergence of a feriet of
positive terms there exists a test for the absolute oonTWgBOee of
any series whatever, real or imaginary. Thus, if the abeeU d e vahie
of the ratio of two consecutive terms of a series |t<,^i/i«,|, after a cer-
tain term, is less than a fixed number less than unity^ the series eon"
verges absolutely. For, let f /", = | m, | . Then, since ( m, ^ , /m. ' < ik < 1
after a certain term, we shall have also
%^<A-<1,
which shows that the series of absolut*- valii«*s
6^0 + ^i + • • • 4- 1. + • •
converges. If |w„^,/w.| approaches a limit I as n becomes if\/lniie,
the series converges if 1<1, and diverges if l>\. The first hmif is
self-evident. In the second case the general term ic, does not
approach zero, and consequently the series (20) and (21) caimoi
both be convergent. The case / = 1 remains in doubt.
More generally, if « be the greatest hmit of -v^ m « becoiMs infinite, Ike
series (19) converges if u<l, and diverges if t0>\. For tn Ibe UU«r CMt Um
modulus of the general term doe« not approach «ero (toe 1 161). The CSH la
which u = 1 remains in doubt — the aeries msj be abK>latel7 ooavsifMt, iteply
convergent, or divergent.
168. Multiplication of aeries. Let
(23) Uo + M, -I- ti, 4- •• -f u.-f •••,
(24)
be any two serit'> .s i ,i; . ; • ^ •:..• !.:•«:
series by terms ol ihr >»... ni ^ .• i; » ■■' ■■ i-**" 1'
852 INFINITE SERIES [VIII, §168
together all the products w,Vy for which the sum i-^j of the sub-
scripts is the same ; we obtain in this way a new series
If each of the series (23) and (24) is absolutely convergent, the
series (25) converges, and its sum is the product of the sums of the
two given series. This theorem, which is due to Cauchy, was gener-
alized by Mertens,* who showed that it still holds if only one of the
series (23) and (24) is absolutely convergent and the other is merely
convergent.
Let us suppose for definiteness that the series (23) converges
absolutely, and let w^ be the general term of the series (25):
The proposition will be proved if we can show that each of the
differences
Wf, + Wi + "- + w^^ -(uo^u^ + ...^u„) (v^ 4_ vi 4- . . . 4- v^),
««'0 + «^l + • • • + tV^n+l - (^0 + t^l + • • • + U, + l)(Vo + ^'l 4- • • • 4- V„ + i)
approaches zero as n becomes infinite. Since the proof is the same
in each case, we shall consider the first difference only. Arranging
it according to the w's, it becomes
8 = "o(V. + I -f • • • + V2n) -h ^l ('^„ + l + • • • + f2«-l) + • • ' + i^n-1^ + 1
+ W, + l(Vo + • • • + V„-l) + U„^^(Vo + • • • + V„_2) + • • • + ^2,1^0-
Since the series (23) converges absolutely, the sum Uo-\-Ui-\ \-U^
is less than a fixed positive number A for all values of n. Like-
wise, since the series (24) converges, the absolute value of the sum
«^o + t^i H h V. is less than a fixed positive number B. Moreover,
corresponding to any preassigned positive number c a number m
exists such that
for any value of p whatever, provided that n>m. Having so chosen
n that all these inequalities are satisfied, an upper limit of the quan-
tity |a{ ii given by replacing u^, u„ w„ • . ., m,. by Uo, U^ f/», • •, ^,.,
• CrtW$ Journal, Vol. LXXIX.
VIII, $ 109] COMPLEX TERBiS MULTIPLE SERIES t6i
respectively, v,^. » 4- v,^, -»-••• + r,^, by ^/{A + B), and finally
of the expressions v© -f vi -f- • • • -f w,_j, v, -f H ».-f ,•••,«»• by JL
This gives
or
|8|<J^(i/o + ^i+-- + t/,-,)-hB(r.n 4- •• + //,.)
whence, finally, | S | < c. Hence the difference 5 actually does i4>proach
zero as n becomes infinite.
169. Double series. Consider a rectangular network which is lim-
ited upward and to the left, but which extends indefinitely down-
ward and to the right. The network will contain an infinite number
of vertical columns, which we shall number from left to right from
to + X . It will also contain an infinite number of horizontal
rows, which we shall number from the top downward from to -f oo.
Let us now suppose that to each of the rectangles of the network a
certain quantity is assigned and written in the corresponding rec-
tangle. Let a,t be the quantity which lies in the »th row and in the
A;th column. Then we shall have an array of the form
(26)
«00
aoi
«os •
• Oo. •••
«.o
a,,
«ij •
• ^l, •••
OlO
<h\
«M
«-0
^m\
**-.«
- ••.,
We shall first suppose that each of the elements of this array is real
and positive.
Now let an infinite sequence of curves Cj, ' . , be drawn
across this array as follows : 1) Any one of them forms with the two
straight lines which Iwund the array a closed curre which entirely
surrounds the preceding one ; 2) The distance from any fixed pdni
to any point of the curve C„ which is otherwise entirely arbitrary,
becomes infinite with n. Let 5, be the sum of the elemenU of the
array which lie entirely inside the closed curve composed of C, and
354 INFINITE SERIES [VIII, § 169
the two straight lines which bound the array. If 5„ approaches a
limit 5 as n becomes infinite, we shall say that the double series
(27) X X«-
1 = k=0
converges f and that its sum is S. In order to justify this definition,
it is necessary to show that the limit 5 is independent of the form
of the curves C. Let C{, Cg, •••, Cl,, ••• be another set of curves
which recede indefinitely, and let 5/ be the sum of the elements
inside the closed curve formed by C{ and the two boundaries. If m
be assigned any fixed value, n can always be so chosen that the
curve C, lies entirely outside of C'^. Hence S'^< S^, and therefore
^L = Sj for any value of m. Since 5^ increases steadily with m, it
must approach a limit S' < 5 as m becomes infinite. In the same
way it follows that S < S'. Hence S' = S.
For example, the curve C^ may be chosen as the two lines which
form with the boundaries of the array a square whose side increases
indefinitely with *, or as a straight line equally inclined to the two
boundaries. The corresponding sums are, respectively, the following :
aoo + (a,oH-aii + aoi)+"-+(a«o + a„i + ----fa„„ + a„-i.„4---- + aon)>
aoo+(aioH-aoi)+(a2o + aii + ao2)+---+Ko + a„_i,i + --- + ao«)-
If either of these sums approaches a limit as n becomes infinite, the
other will also, and the two limits are equal.
The array may also be added by rows or by columns. For, sup-
pose that the double series (27) converges, and let its sum be S. It
is evident that the sum of any finite number of elements of the series
cannot exceed S. It follows that each of the series formed of the
elements in a single row
(28) a,o -}- a.i + • • • + «,n + • • • , i = 0, 1, 2, • • • ,
converges, for the sum of the first n+1 terms a^Q + a,i H h *•»
cannot exceed S and increases steadily with n. Let o-, be the sum of
the series formed of the elements in the tth row. Then the new series
surely converges. For, let us consider the sum of the terms of the
array Sa^^ for which t</>, A;<r. This sum cannot exceed S, and
increases steadily with r for any fixed value of jo; hence it
approaches a limit as r becomes infinite, and that limit is equal to
(^^) <^» + <r, + • • • + CTp
I
vni,§l«9] COMPLEX TER>f- 'ILTIPLE SERIES 856
for any fixed value of p. It foUow8 that <r« + cti 4- . • 4. ^^ '^^ rr ^
exceed S and increases steadily with p. Consequently the seriet (29)
converges, and its sum S is less than or equal to .V. Coov«nely, if
eacii of the series (28) converges, and the series (29) oonverget to a
sum :^, it is evident that the sum of any finite number of elamente
of the array (26) cannot exceed 2. Hence S<X and ooDMqiiaiitlj
The argument just given for the series formed from the elements
in individual rows evidently holds equally well for the aeriea formed
from tiie elements in individual columns. The sum of a ftimwufiMf
double series whose elements are all positive may be evalmated ky
roivSf by columnsy or by means of curves of any form which rectie
indefinitely. In particular^ if the series eowBergea when added by tmoe^
it will surely converge when added by columns^ and the eum will be the
same. A number of theorems proved for simple series of positiYe
terms may be extended to double series of positive elements. For
example : if each of the elements of a double series of positive elewiemte
is lesSf respectively, than the correeponding elements of a known eon'
vergent double series, the first series is also convergent; and ao forth.
A double series of positive terms which is not convergent is said
to be divergent. The sum of the elements of the corresponding
array which lie inside any closed curve increases beyond all limit
as the curve recedes indefinitely in every direction.
Let us now consider an array whose elements are not all positire.
It is evident that it is unnecessary to consider the cases in which
all the elements are negative, or in which only a finite number of
elements are either positive or negative, since each of these eases
reduces immediately to the preceding case. We shall therafore sup-
pose that there are an infinite number of positive elements and an
infinite number of negative elements in the array. Let a^ be the
general term of this array T. If the array T, of positiTe alemsnta,
each of which is the absolute value \a^^\ of the corresponding elsoiant
in Ty converges, the array T is said to be absolutely eemperyenL Snob
an array has all of the essential properties of a oooTergant array of
positive elements.
In order to prove this, let us consider two auxiliary arrays T
and r\ defined as follows. The array T* is formed from the array T
by replacing each negative element by a lero, retaining the positive
elements as they stand. Likewise, the array 7^ is obtained from
the array T by replacing each positive element by a sero and chang-
ing the sign of each negative element Each of the arrays T and 7*
866 INFINITE SERIES [VIII, §169
converges whenever the array Ti converges, for each element of 7^,
for example, is less than the corresponding element of Tj. The sum
of the terms of the series T which lie inside any closed curve is
equal to the difference between the sum of the terms of V which
lie inside the same curve and the sum of the terms of r" which
lie inside it. Since the two latter sums each approach limits as
the curve recedes indefinitely in all directions, the first sum also
approaches a limit, and that limit is independent of the form of
the boundary curve. This limit is called the sum of the array T.
The argument given above for arrays of positive elements shows
that the same sum will be obtained by evaluating the array T by
rows or by columns. It is now clear that an array whose elements
are indiscriminately positive and negative, if it converges absolutely j
may be treated as if it were a convergent array of positive terms.
But it is essential that the series Tj of positive terms be shown to
be convergent
If the array Ti diverges, at least one of the arrays T and T' diverges. If
only one of them, T for example, diverges, the other T'" being convergent, the
sum of the elements of the array T which lie inside a closed curve C becomes
infinite as the curve recedes indefinitely in all directions, irrespective of the
form of the curve. If both arrays T and T' diverge, the above reasoning
•howi only one thing, — that the sum of the elements of the array T inside
a closed curve C is equal to the difference between two sums, each of which
Increaaes indefinitely as the curve C recedes indefinitely in all directions. It
may happen that the sum of the elements of T inside C approach different
limits according to the form of the curves C and the manner in which they
recede, that is to say, according to the relative rate at which the number of
positive terms and the number of negative terms in the sum are made to increase.
The sam may even become infinite or approach no limit whatever for certain
methods of recession. As a particular case, the sum obtained on evaluating by
rows may be entirely different from that obtained on evaluating by columns if
the array is not absolutely convergent.
The following example is due to Amdt.* Let us consider the array
(SI)
2(2/ "sU'sls) "4(4) '.'••' ;(V") "FTi(?Tl)'
2(2} "iW'sls) "4(4)*--'jp(^) -jilfe)*
• Orunert'B ArcMv, Vol. XI, p. 319.
VIII. fie»] COMPLKX TER1C8 MULTIPLE HER1K8 367
which contains an infinite number of poaicife ao<l aa
elementii. Each of the serien foraad from tte alMBMrti in a
those in a single column coQTWfia. Tha ram of tha
terma in the nth row ia aTtdantlgr
i/iV JL
Hence, evaluating the array (81) by rowa, the reaoli obuUnad Im aqoal to tha
sum of the convergent seriea
which ifl 1/2. On the other hand, the aariaa formed from the elemniili In tha
(p — l)th column, that is,
converges, and its sum is
P P+1 P(P+1) P + 1 P*
Hence, evaluating the array (31) by columns, the reanlt obtained la equal to the
sum of the convergent series
(i-i)*(:-i)--(;^-j)
which is — 1/2.
This example shows clearly that a double aerlee ahonld not be need ia a
calculation unless it is absolutely convergent.
We shall also meet with double series whose elements are complex
quantities. If the elements of the array (26) are complex, two oihar
arrays V and T' may be formed where each element of 7* ia th«
real part of the corresponding element of Tand each element of 7**
is the coefficient of t in the corresponding element of 7*. If the
array T^ of absolute values of the elements of T, each of whoae
elements is the absolute value of the corresponding element of T,
converges, each of the arrays T and T" oonverges abiolntaly, and
the given array T is said to be absoluUlif eomn^rgenL The sum of
the elements of the array which lie inside a Tariable cloaed cunre
approaches a limit as the curve recedes indefinitely in all directions.
This limit is independent of the form of the Tariable corre, and it
is called the sum of the given array. The sum of any absolutet^
convergent array may also be evaluated by rows or by colimins.
858 INFINITE SERIES [VIII, §170
170. An absolutely convergent double series may be replaced by a simple
seriM formed from the same elements. It will be sufficient to show that the
roctangles of the network (26) can be numbered in such a way that each rec-
tangle ha» a definite number, without exception, different from that of any other
roctangle. In other words, we need merely show that the sequence of natural
nambeFB
(82) 0, 1, 2, ..., n, •.-,
and the assemblage of all pairs of positive integers (i, k), where i>0, A;>0, can
be paix«d oil in such a way that one and only one number of the sequence (32)
will correspond to any given pair (i, fc), and conversely, no number n corresponds
to more than one of the pairs (i, A:). Let us write the pairs (i, k) in order as
follows :
(0,0), (1,0), (0,1), (2,0), (1,1), (0,2), ...,
where, in general, all those pairs for which i -h k = n are written down after
those for which i + k<n have all been written down, the order in which those
of any one set are written being the same as that of the values of i for the various
pairs beginning with (n, 0) and going to (0, n). It is evident that any pair (i, k)
will be preceded by only & finite number of other pairs. Hence each pair will
have a distinct number when the sequence just written down is counted off
according to the natural numbers.
Suppose that the elements of the absolutely convergent double series SSoifc are
written down in the order just determined. Then we shall have an ordinary series
(33) ooo + aio + Ooi + aao + an + aoz H + ^no + On-i.i H
whose terms coincide with the elements of the given double series. This simple
series evidently converges absolutely, and its sum is equal to the sum of the given
double series. It is clear that the method we have employed is not the only pos-
sible method of transforming the given double series into a simple series, since
the order of the terms of the series (33) can be altered at pleasure. Conversely,
any absolutely convergent simple series can be transformed into a double series
in an infinite variety of ways, and that process constitutes a powerful instrument
in the proof of certain identities.*
It Lb evident that the concept of double series is not essentially different from
that of simple series. In studying absolutely convergent series we found that
the order of the terms could be altered at will, and that any finite number of
terms could be replaced by their sum without altering the sum of the series.
An attempt to generalize this property leads very naturally to the introduction
of double series.
171. Multiple series. The notion of double series may be generalized.
In the first place we may consider a series of elements a„,„ with two
subscripts m and n, each of which may vary from — oo to -|- oo .
The elements of such a series may be arranged in the rectangles of
a rectangular network which extends indefinitely in all directions ;
•Tannery. Tntroduetion a la thtforie des fonctions d'une variable, p. 67.
Viii, §172] COMPLEX TERMS M CLT J PLE SERIES S59
ii 1.^ evident that it may be divmea into four double Mriet of tht
type we have just studied.
A more important generalization is the following. Let ut
a series of elements of the type a^,^,,,^ where the tul
TKi, TTi^f "t rn^ may take on any values from to + oo, or from — oe
to + 00, but may be restricted by certain inequalitiea. Although no
such convenient geometrical fonn as that used above if ATailable
when the number of subscripts exceeds three, a slight oonsideratioo
shows that the theorems proved for double series admit of immediate
generalization to multiple series of any order p. Let us first sup-
pose that all the elements «„,,.„,,...,,« are real and positive. Let Si
be the sum of a certain number of elements of the given seriee, ^
the sum of .s\ and a certain number of terms previously neglected,
Ng the sum of .s'2 and further terms, and so on, the successive sums
.Si, S^y • •, N„, •• being formed in such a way that any particular
elemcnt of the given series occurs in all the smns past a certain one.
If S^ approaches a limit S as ti becomes infinite, the given seriee
is said to be convergent^ and S is called its sum. As in the case of
double series, this limit is independent of the way in which the
successive sums are formed.
If the elements of the given multiple series have different signs
or are complex quantities, the series will still surely converge if the
series of absolute values of the terms of the given series ooDTeiget.
172. Generalization of Cauchy's theorem. The following theotem,
which is a generalization of Cauchy's theorem (§ 161), enables us to
determine in many cases whether a given multiple series is conver-
gent or divergent. Let /(a-, y) be a function of the two variablet x
and y which is positive for all points (x, y) outside a certain closed
curve r, and which steadily diminishes in value as the p« •/)
recedes from the origin.* Let us consider the value of t: . io
integral///(a;, y) dx dy extended over the ring-shaped region
r and a variable curve C outside T, which we shall allow to
indefinitely in all directions ; and let us compare it with the double
series 2/(m, n), where the subscripts m and n may assume any posi-
tive or negative integral values for which the point (w, n) lies out^
side the fixed curve T. Then the doubU mrim oemoergu iftks
integral approaches a limit f and convenely.
• All that is necessary for the present proof b UuU/(«i, |fi)^/(«^. Wd
zi>z, and Vx'^Vt outside T. It Is easy to adapt the proof to rtOl bm
hypotheses. — TaAMs.
860 INFINITE SERIES [vm,§m
The lines a; = 0, x = ±1, a; = ± 2, • • andy = 0, y = ±1, y = ±2, • ••
divide the region between r and C into squares or portions of squares.
Selecting from the double series the term which corresponds to that
comer of each of these squares which is farthest from the origin, it
is evident that the sum 2/(w, n) of these terms will be less than the
value of the double integral Jff(xj y) dx dy extended over the region
between r and C. If the double integral approaches a limit as C
leoedes indefinitely in all directions, it follows that the sum of any
number of terms of the series whatever is always less than a fixed
number ; hence the series converges. Similarly, if the double series
converges, the value of the double integral taken over any finite
region is always less than a fixed number; hence the integral
approaches a limit. The theorem may be extended to multiple
series of any order jo, with suitable hypotheses; in that case the
integral of comparison is a multiple integral of order p.
As an example consider the double series whose general term is
l/(m* -\- w*)**, where the subscripts m and n may assume all integral
values from — oo to + oo except the values m = n = 0. This series
converges for /ti > 1, and diverges for ^^1. For the double integral
<»» ff<^>
extended over the region of the plane outside any circle whose
center is the origin has a definite value if /a > 1 and becomes
infinite if /i<l (§133).
More generally the multiple series whose general term is
1
(mJ + mi + '-' + mJ)'*'
where the set of values mi = mg = • • • = mp = is excluded, con-
verges if 2/A > p.*
III. SERIES OF VARIABLE TERMS UNIFORM CONVERGENCE
173. Definition of uniform convergence. A series of the form
(35) «o(aj)+ Mi(a;)-f • • + t^n(«)+ •••,
whose terms are continuous functions of a variable a; in an inter-
ral (a, 6), and which converges for every value of -^ belonging to
that interval, does not necessarily represent a continuous function,
UMorwna are to be found in Jordan's Cours d 'Analyse, Vol. I, p. 163.
VIII, $173] VARIABLE TERMS 861
as we might be tempted to believe. In order to prore the fael W9
need only consider the series studied in | 4 :
which satisfies the above conditions, but whose sum is di«oontinuous
for X = 0. Since a large number of the functions which occur in
mathematics are defined by series, it has been found neceeiary to
study the properties of functions given in the form of a series. The
first question which arises is precisely that of determining whelber
or not the sum of a given series is a continuous function of the
variable. Although no general solution of this problem is known,
its study has led to the development of the very important notion
of uniform converyence.
A series of the type (35), each of whose terms is a function of x
which is defined in an interval (a, 6), is said to be w»tforml}f eam^
vergent in that interval if it converges for every value of x between
a and b^ and if, corresponding to any arbitrarily preassigned positive
number e, a positive integer JV, independent of or, can be found such
that the absolute value of the remainder R^ of the given aeries
^* = w.+i(-^)+ ".+«(«)+••• + •*.+,(«)+ •••
is less than « for every value of n>N and for every value of »
which lies in the interval (a, b).
The latter condition is essential in this definition. For any pre-
assigned value of x for which the series converges it is apparent
from the very definition of convergence that, corresponding to any
positive number €, a number N can be found which will satisfy
the condition in question. But, in order that the series should con-
verge uniformly, it is necessary further that the same number N
should satisfy this condition, no matter what value of x be selsoted
in the interval {a, b). The following examples show that soeh is not
always the case. Thus in the series considered just above we have
The series in question is not uniformly convergent in the inter-
val (0, 1). For, in order that it should be, it would be «eeesMrjf
(though not sufficient) that a number N exist, such that
1 ^
(1-hx*)'
862 INFINITE SERIES [VllI, § 173
for all values of x in the interval (0, 1), or, what amounts to the
same thing, that
1 -f a:^ > e^
1 1 1
Whatever be the values of N and e, there always exist, however,
positive values of x which do not satisfy this inequality, since the
right-hand side is greater than unity.
Again, consider the series defined by the equations
5.(x)=nxe-^, S,{x)=^, u,(x) = S, - S„_,, w = l,2, ....
The sum of the first n terms of this series is evidently S,^ (x), which
approaches zero as n increases indefinitely. The series is therefore
convergent, and the remainder Rn{x) is equal to — wxe""^. In order
that the series should be uniformly convergent in the interval (0, 1),
it would be necessary and sufficient that, corresponding to any arbi-
trarily preassigned positive number c, a positive integer N exist such
that for all values of n>N
nxe-'^<€, 0<a;<l.
But, if a; be replaced by l/?i, the left-hand side of this inequality is
equal to e"^/", which is greater than 1/e whenever n> 1. Since c
may be chosen less than 1/e, it follows that the given series is not
uniformly convergent.
The importance of uniformly convergent series rests upon the
following property:
The sum of a series whose terms are continuous functions of a
variable x in an interval (a, h) and which converges uniformly in that
intervalf is itself a continuous function ofx in the same interval.
Let Xo be a value of x between a and ft, and let Xq-\- hhQ a value
in the neighborhood of Xq which also lies between a and b. Let n
be chosen so large that the remainder
^«(«) = ^«+i(a5) + w„+2(a;) -h • • •
is less than c/3 in absolute value for all values of x in the interval
(tf, b), where c is an arbitrarily preassigned positive number. Let/(a;)
be the sum of the given convergent series. Then we may write
/(x)=^(x)-h/e,(x),
where 4i(x) denotes the sum of the first w -f 1 terms,
^{x) = M„(x) 4- w, (x) 4- . . . + w,(x).
VllI, §m] VARIAHLE TPUM8 $5$
Subtracting the two equalities
/(a^)=0(x,)-fi?.(x,),
f{x, 4- A) = ^(x, + A) + i2.(«, 4. A),
we find
fix, 4- h) ^f(x,) = [^(xo 4- A) - ^(x,)] 4- i?.(x, 4- A) - i?.(x,).
The number 71 was so chosen that we have
On the other hand, since each of the terms of the series is a oontintt-
ous function of x, <f>(x) is itself a continuous function of x. Heoee
a positive number rj may be found such that
\^x, + h)-4K.x,)\<l
whenever \h\ is less than rj. It follows that we shall hare, a fortiori,
lA^o-f /o-yi*o)i<3|
whenever | ^ | is less than 1;. This shows that f(x) is cootinuoos
for x = Xq.
Note. It would seem at first very difficult to determine whether
or not a given series is uniformly conver^nt in a given inteml.
The following theorem enables us to show in many cases that a
given series converges uniformly.
Let
(36) Wo(ar)4- u»(x)4- ••• + «,(«) +
be a series each of whose terms is a amHnuoms /knetion qf x in mn
interval (a, b), and let
(37) 3/,4.3/j4---- + 3r,4---
be a convergent series whose terms are positive oomstamts. Thorn,
(/* I M, I < 3/, for all values of x in the interval (a, h) and for ali
values of n, the first series (36) eonver^ umtforwdjf in the i m t ero mi
considered.
For it is evident that we shall have
|w. + i + M.^,4----|<iCi^,4-^,^,-»-'--
864 INFINITE SERIES [VIII, §174
for all values of x between a and h. If JV be chosen so large that
the remainder H^ of the second series is less than c for all values
of n greater than iV, we shall also have
whenever n is greater than iV, for all values of x in the interval (a, h).
For example, the series
3/o -h 3f 1 sin x -\- M^ sin 2x H V M^ sin nx -\- •",
where 3/o, A/i, 3/,, ••• have the same meaning as above, converges
uniformly in any interval whatever.
174. Integration and differentiation of series.
Any series of continuous functions which converges uniformly in an
inter acU (a, b) may be integrated term by term, provided the limits of
integration are finite and lie in the interval (a, b).
Let Xq and Xi be any two values of x which lie between a and b,
and let jV be a positive integer such that \R^{x)\<€ for all values
of X in the interval (a, b) whenever n> N. Let f(x) be the sum of
the series
f{x) = Uo(x) + wi (x) + • • • + i^„(ic) + • • •,
and let. us set
^.= 1 f{x)dx-j Uodx- I uidx / \^dx=( ^R^dx.
The absolute value of D, is less than €\xi—Xo\ whenever n>N.
Hence Z). approaches zero as n increases indefinitely, and we have
the equation
/ f{x)dx=l Uo(x)dx-{- ui(x)dx + -..i- f \(x)dx + --..
*''• *^*% *>'*o Jx^
Considering x^ as fixed and x^ as variable, we obtain a series
J Uo(x)dx + -'.-^ j u^(x)dx + ...
which converges uniformly in the interval (a, b) and represents a
oondDuout function whose derivative is f{x).
VI", §174] VARIABLE TER>IS
Conversely, any eanverffent series may be d^emUiaied term hy term
if the resulting series converges untformly*
For, let
f(x) a tio(«) + tt|(x) + •• • + ti.(x) + ...
be a series which converges in the interval (a, h). Let us suppose
that the series whose terms are the derivatives of the terms of the
given series, respectively, converges uniformly in the tame ioterral,
and let </>(x) denote the sum of the new series
Integrating this series term by term between two limits jr« aod a^
each of which lies between a and 6, we find
I
<i>{x)dx = [uo(ar) - Uo(Xo)] + [tt|(x) - tti(av)] +
or
f <l>(x)dx=/(x)^/(x.).
This shows that <^(x) is the derivative oi fXz\.
Examples. 1) The integral
/
X
cannot be expressed by means of a finite number of elemsntaiy
functions. Let us write it as follows:
The last integral may be developed in a series which holds for all
values of x. For we have
1.2 ■ 1.2.3 ' ■ 1.2 n
and this series converges uniformly in the interval from — i? to -f i?,
no matter how large R be taken, since the absolute Talus of aoj
* It is assumed in the proof also that aaeh tonn of tiM mw
(unctioD. The theorem ia true, however, la g wi i ml. ~ TaAim.
866 INFINITE SERIES [viii, §174
term of the series is less than the corresponding term of the con-
vergent series
It follows that the series obtained by term-by-term integration
X 1 X* 1 x^
converges for any value of x and represents a function whose deriva-
tive is (^ — l)/x.
2) The perimeter of an ellipse whose major axis is 2a and whose eccentricity
is e is equal, by § 112, to the definite integral
w
S = 4a r* VI - e2sin20d0.
The product c^sin^ lies between and ^{<1). Hence the radical is equal to the
turn of the series given by the binomial theorem
VI - e^sin^^ = 1 - - e^sin*^ e*sin*0
2 2.4
1.3.5-..(2n-3) , . ,
2 . 4 . 6 • . . 2n
The series on the right converges uniformly, for the absolute value of each of
its terms is less than the corresponding term of the convergent series obtained
by setting sin^ = 1. Hence the series may be integrated term by term; and
since, by § 116,
r ^ sin*" <p d<t>
Jo
5«.^. 1.3.6...(2n-l) ;r
2 . 4 . 6 . . . 2n 2
shall have
w
rVl-c«sin«0(i0 = -(l-lc2-Ae*-Ae6
Vo 2 ( 4 64 256
rl.8.5-.-(2n-3n2 ^
-[ 2.4.6...2n ^ ](2n-l)e^"--}.
If the eccentricity e is small, a very good approximation to the exact value of the
integral is obtained by computing a few terms.
Similarly, we may develop the integral
r Vl-e"^sin2 0d0
Jo
in a series for any value of the upper limit <f>.
Finally, the development of Legendre's complete integral of the first kind
leads to the formula
*
r-—!f^ ;{. + I^+ »,.^ ■ I r l.8.6-.-(2»-l) -|' I
J, Vl-CIn'* 2} ^4 ^84 ^ *L 2.4.6...2« J + j"
VIII. $174] VARIABLK TERMS %$^^
The definition of uniform convergence way be extended to lene*
whose terms are functions of several independent variablet. For
example, let
Wo i^f y) + W| (x, y) 4- • + M, (X, y) + •
be a series whose terms are functions of two independent Tarimblet x
and y, and let us suppose that this series converges whenever tht
point (x, y) lies in a region R bounded by a oloeed eootoor C,
The series is said to be uniformly convergent in the regioo R ML
corresponding to every positive number c, an integer N can be found
such that the absolute value of the remainder R^ is lees than c
whenever n is equal to or greater than Nj for every point (;r, y)
inside the contour C. It can be shown as above that the sum of
such a series is a continuous function of the two variables x and
y in this region, provided the terms of the series are all continu-
ous in R.
The theorem on- term-by-term integration also may be generalixed.
If each of the terms of the series is continuous in R and if /(x, y)
denotes tlie sum of the series, we shall have
\ J f{^iy)dxdy= j j u^{x,y)dxdy'^j j u^{x,y)dxdy '\- -.
^J J ^^n{x>y)dxdy{'"-,
where each of the double integrals is extended over the whole Inte-
rior of any contour inside of the region R.
Again, let us consider a double series whose elements arw tunruoos
of one or more variables and which converges absolutely for all seta
of values of those variables inside of a certain domain D. Let the
elements of the series be arranged in the ordinary rectangular amy,
and let R^ denote the sum of the double series outside any eloeed
curve C drawn in the plane of the array. Then the given double
series is said to converge uniformly in the domain D if oomspood-
ing to any preassigneil number c, a closed curve i?, not depaodeni
on the values of the variables, can be drawn such that \R,\<t for
any curve C whatever lying outside of K and for any set of values
of the variables inside the domain D.
It is evident that the preceding de6nitions and tbeorams may be
extended without difficulty to a multiple series of any order
elements are functions of any number of yariablea.
868 INFINITE SERIES [VIII, §176
NoU. If a aeries does not converge uniformly, it is not always allowable to
integrate it term by term. For example, let us set
fl;(x) = iU5e-'«*, 5o(x) = 0, u^(x) = S„-5„_i. n = l, 2, ....
The series whose general term is Un (x) converges, and its sum is^ero, since Sn (x)
si^KYWches sere as n becomes infinite. Hence we may write
/(x) = = ui(x) + 1I2(X) + '. . + M«(x) + . .,
whence Sq/(x) dx = 0. On the other hand, if we integrate the series term by
term between the limits zero and unity, we obtain a new series for which the
sum of the first n terms is
/»1 T-p-nx'-il 1
j;s.(x)d» = -[— ]^=-(l-e-»),
which approaches 1/2 as its limit as n becomes infinite.
175. Application to differentiation under the integral sign. The proof
of the formula for differentiation under the integral sign given in
§ 97 is based essentially upon the supposition that the limits Xq
and X are finite. If X is infinite, the formula does not always hold.
Let us consider, for example, the integral
sin ax
ax, a > 0,
This integral does not depend on a, for if we make the substitu-
tion y = ax it becomes
[ siny
y
If we tried to apply the ordinary formula for differentiation to F(a),
we should find
^'^=£
F^(a) = j cos ax dx
This is surely incorrect, for the left-hand side is zero, while the
right-hand side has no definite value.
Sufficient conditions may be found for the application of the
ordinary formula for differentiation, even when one of the limits
is infinite, by connecting the subject with the study of series. Let
u« first consider the integral
X
f(x)dx,
whieh we shall suppose to have a determinate value (§ 90). Let
•i, a,, ..., a,, .. ■ be an infinite increasing sequence of numbers, all
VIII, (m] VARIABLE TBRlfg 359
greater than Oo, where a, becomea inlmite with it. If we aei
U.^J\x)dx, U,=J\x)dx, ..., U.^J'^'*/{g)du,
the series
converges and its sum is /^* *f(^jr) rfar, for the sum A\ of the finrt n termi
is equal to f^f(x)dx.
It should be noticed that the converse is not always true.
If, for example, we set
/(x) = C08x, 00=0, ai = 7r, «. = »«•, -,
we shall have
%/nn
COS xdx ssO,
Hence the series converges, whereas the integral f'ooBxdx ap-
proaches no limit whatever as / becomes infinite.
Now let f{x, a) be a function of the two variables x and a which
is continuous whenever x is equal to or greater than <^ and a lies
in an interval (a©, ai). If the integral J'/{x, a)dx approaches a
limit as I becomes infinite, for any value of a, that limit is a
function of a,
FL.\=\ /{x,a)dx,
'\
which may be replaced, as we have just shown, by the sum of a
convergent series whose terms are continuous functions of a:
F(a) = Uo(a) + f/^ (a) -f • • 4- U,(a) + ••.,
This function F(a) is continuous whenever the series copf w g M ani-
formly. By analogy we shall say that ths integral {^* J{x^ a) dx
converges uniformly in the interval (a©, <ti) if, oorrespoiidiiig to any
preassigned positive quantity c, a number ^ independent of a emu
be found such that | Jj"^*/!;*, a)dx\<% whenever /> S, for a^y value
of a which lies in the interval («•, at).* If tlie integnd oonwfti
• See W. F. Osgood. AwmU <(f M ath sm o Hm , M Mtkt, Vol. HI 0«H). F im —
Trams.
370 INFINITE SERIES [VIII, §176
unifonnly, the series will also. For if a„ be taken greater than N,
we shall have
I f(x,a)dx
<€;
henoe the function F{a) is continuous in this case throughout the
intenral (a©, ai).
Let us now suppose that the derivative df/da is a continuous
function of x and a when x^a© and aQ<a^aij that the integral
X
f-dx
Ca
has a finite value for every value of a in the interval (aro, o-j), and
that the integral converges uniformly in that interval. The integral
in question may be replaced by the sum of the series
X
where
The new series converges uniformly, and its terms are equal to the
corresponding terms of the preceding series. Hence, by the theorem
proved above for the differentiation of series, we may write
n<^)=£'Yj^.
In other words, the formula for differentiation under the integral sign
still holds, provided that the integral on the right converges uniformly.
The formula for integration under the integral sign (§ 123) also
may be extended to the case in which one of the limits becomes
infinite. Let /(a*, a) be a continuous function of the two variables
X and a, for x>aQ,aQ<a<a^. If the integral S^"^ f{x, a) dx is uni-
formly convergent in the interval (ao, «i), we shall have
I dx f f(x,a)da= f da I f(x,a)dx.
To prove this, let us first select a number l> a^^ then we shall
hare
(B) J ^f '/(«' a)da = j 'da Cf(ic, a) dx ,
VIII. §176] VARIABLE TERMS S7l
As / increases indefinitely the right-hand side of tkii
approaches the double integral
da j JXx,a)dXt
ioi the difference between these two double integrmlB is equal to
f{z, a)dx.
SH
Suppose N chosen so large that the absolute value of the integral
ii^^fi^y oc)dx is less than c whenever I is greater than A', for any
value of a in the interval (ao, a^). Then the absolute value of the
difference in question will be less than c|ai — a^\^ and therefore it
will approach zero as I increases indefinitely. Henoe the left-hand
side of the equation (B) also approaches a limit as / becomes in6-
nite, and this limit is represented by the symbol
J /»■!-• /»«,
I dx \ f{x,a)da.
This gives the formula (A) which was to be proved.*
176. Examples. 1) Let us return to the int^ral of f 01 :
sinz
'""=r
X "'
where a is positive. The integral
-J r-^sinxdz,
• The formula for difTerentiation may be deduced easily tram t^ foraiala (A). Fbr.
suppose that the two functions /(x, a) aiid/«(z. a) are contlBOoas for a^<a<a\,
a; > ao ; that the two integrals F(a) = j^ *f{z, a)d*noA ♦(a) = ^ "/, (». a) d» hare
finite values ; and that the latter conrergee oniformly in the loU«^ (a^, aj). Froai
the formula (A), if (r lies in the intenral (aoi cci), we have
where for distinctness a has been replaced by « under tba lalsimU liffa. Mt lUs
formula may be written In the form
f'*{u) dtt = r * */(x, a)dx-' f* '/(«, a«)ds « tia) - r(a^ ,
whence, taking the derivattre of each side with raspeol to a, wa ted
rCa) = ♦(«).
372 INFINITE SERIES [VIII, §176
obtained by differentiating under the integral sign with respect to a, converges
uniformly for all values of a greater than an arbitrary positive number k. For
we have
U + oc I /»+<» \
e-«*sinxdx </ e-'^'^dx = —er^i ,
Mid hence the absolute value of the integral on the left will be less than e for all
values of a greater than k,ifl> N^ where N is chosen so large that A:e*^ > 1/e
It follows that
e-'^^sinxdlx.
"«') = [-
The indefinite integral was calculated in § 119 and gives
■e-<»'*(cosa; + arsinx)l+* _ —1
l+a:2 Jo ~ l+a^'
whence we find
F{a) = C — arc tan a,
and the constant C may be determined by noting that the definite integral F{a)
approaches zero as a becomes infinite. Hence C = 7r/2, and we finally find the
formula
X
sinx , ,1
e- «* dx = arc tan — .
z a
This formula is established only for positive values of a, but we saw in § 91 that
the left-hand side is the sum of an alternating series whose remainder E„ is always
less than 1/n. Hence the series converges uniformly, and the integral is a con-
tinuous function of a, even for a = 0. As a approaches zero we shall have in
the limit
r
/on\ I Sm X , 7t
(8») I ^'^=2
2) If in the formula
Jo 2
of { 184 we set X = y Va, where a is positive, we find
(40) , J^"'V«v.dy = :^a-i,
and it is easy to show that all the integrals derived from this one by successive
differentiations with respect to the parameter a converge uniformly, provided
that a is always greater than a certain positive constant k. From the preceding
formula we may deduce the values of a whole series of integrals :
(«)
r
t/o
V^e-^^dy^l^V^a-i.
y2 e-''i^dy = ^ a'
22
f;'^.e-^,y = hl-^.^^^I^^,„-'-^.
VIU, Eu] KXERCI8ES 171
By combining iheHe an infinite number of other totafimla may bt •valaated.
We have, for example,
All the integralB on the right have been evalaated above, and we IIimI
1 fx (2/3)« Va a- 1
Jo 2 \a 1.2 2 2
+(-iy
1.2.3. -^n 2
or, simplifying,
2 Ircr
EXERCISES
1. Derive the formula
«■ V* ''3.s...(«»~i) ^.iyi
1 il [X- (logx)-] = 1 + Si log* + ;% (logz)« + . . . + r-p— (>«««)••
1.2..ndx''"'°'" " 1.2^^' I.2...«
where iSp denotes the sum of the products of the first n natural numbers taken f
at a time. [Muamv.)
[Start with the formula
and differentiate n times with respect to x.]
2. Calculate the value of the definite integral
X
'0
by means of the formula for differentiation onder Um Inlafral sign.
3. Derive the formula
[First show that dl/da = - 2/.]
374 INFINITE SERIES [Viii, Ex&
4. Derive the formula
Jo va
by making use of the preceding exercise.
6. From the relation
derive the formula
1 1 /* + *
a« 2 Jo
CHAPTER IX
POWER SERIES TRIGONOMETRIC SERIES
In this chapter we shall study two particularly important
of series — power series and trigonometric serien. Although we ihall
speak of real variables only, the arguments used in the study of
power series are applicable without change to the case where the
variables are complex quantities, by simply substituting the expree-
sion inodulHn or absolute value {pt a complex variable) for the ex p ree*
8 ion absolute value (of a real variable).*
T. POWER SERIES OF A SINGLh >AiuA»LE
177. Interval of convergence. Let us first consider a series of the form
(1) ^lo 4- A^X + A^X} + • • • + vl.-Y- + • ••,
where the coefficients >lo> ^i, ^«, ••• are all positive, mod where
the independent variable A' is assigned only positive values. It it
evident that each of the terms increases with -V. Hence, if the
series converges for any particular value of A, say A'l , it oonvergei
afoHiori for any value of X less than A'j. Conversely, if the series
diverges for the value A',, it surely diverges for any vmlue of X
greater than A,. We shall distinguish tlie following cas e s .
1) The series (1) may converge for any value of A whatever.
Such is the case, for example, for the series
Y Y* A"
1 ^1.2 ' 1.2 -n
2) The series (1) may diverge for any value of X except X —
The following series, for example, has this property:
l + A+1.2A«-h...+1.2.3..fiA»+ .
3) Finally, let us suppose that the series oonvergt« f«.r r«»rtAio
values of A and diverges for other values. Let A, be a valuo of X
for which it converges, and let Ag ^ » ^*lu« 'o' which it diverges.
•See Vol. \l,^WMa^.^'tm^Jn,
876
376 SPECIAL SERIES [IX, §177
From the remark made above, it follows that Xi is less than Xg. The
series converges if A'<A'i, and it diverges if A>Aa. The only
uncertainty is about tlie values of X between Aj and X^ . But all
the values of A for which the series converges are less than A2, and
hence they have an upper limit, which we shall call R. Since all the
values of A for which the series diverges are greater than any value
of A for which it converges, the number R is also the lower limit of
the values of A for which the series diverges. Hence the series (1)
diverges for all values of X greater than Ry and converges for all values
of X less than R. It may either converge or diverge when X = R.
For example, the series
l + A + A2-|-...+A» + ..-
converges if A < 1, and diverges if A^ ^ 1. In this case R =1.
This third case may be said to include the other two by suppos-
ing that R may be zero or may become infinite.
Let us now consider a power series, i.e. a series of the form
(2) tto + aix + a^x^ -\ 1- «„«» -\ ,
where the coeflScients a, and the variable x may have any real values
whatever. From now on we shall set .4,- = |a,.|, Z = |a;|. Then the
series (1) is the series of absolute values of the terms of the series (2).
Let R be the number defined above for the series (1). Then the
series (2) evidently converges absolutely for any value of x between
— R and -f- R, by the very definition of the number R. It remains
to be shown that the series (2) diverges for any value of x whose
absolute value exceeds R. This follows immediately from a funda-
mental theorem due to Abel : *
If the series (2) converges for any particular value Xq, it converges
ahsolutely for any values ofx whose absolute value is less than \xq\.
In order to prove this theorem, let us suppose that the series (2)
converges for x = Xq, and let 3/ be a positive number greater than
the absolute value of any term of the series for that value of x.
Then we sliall have, for any value of n,
md we may write
i4,A*
^•|''Kifi)"<"fe)"
• ifecA«rcA« «tir /o «fr<a 1 + ^ X + ^^?^^^^?-lH xa +
IX, §177] POWER 8ERIB8 177
It follows that the series (1) conyerges whenerer X<\a^\, whlah
proves the theorem.
In other words, if the series (2) converges for z « z«, the seriat (1)
of absolute values converges whenever X is lest than Ijb^I. Heaoo
I^Tol caiinot exceed R, for R was supposed to be the upper limit of
the values of X for which the series (1) converges.
To sum up, given a power series (2) whose ooefficients maj ha¥«
either sign, there exists a positive number R which has the follow-
ing properties : The series (2) converges absolutely for any value of x
between — R and -f- R, and diverges for any value of x whose nhmluie
value exceeds R. The interval (— Rj -^ R) is called the iniervai of
convergence. This interval extends from — oo to + oo in the case in
which R is conceived to have become infinite, and reduoet to the
origin ii R = 0. The latter case will be neglected in what follows.
The preceding demonstration gives us no information about what
happens when x = R or x = — R. The series (2) maj be absolutalj
convergent, simply convergent, or divergent For example, /? = 1
for each of the three series
l-\-x -f-x« + ... + x" + ..-,
X X 3?*
! + _ + _ + . .. + _ + ...,
for the ratio of any term to the preceding approaches x as its limit
in each case. The first series diverges for z = ± 1. The seoood
series diverges for x = 1, and converges for x = — 1. The third con-
verges absolutely for x = ±1.
Note. The statement of AbePs theorem may be made more geoaial,
for it is sufiicient for the argument that the absolute value of aoj
term of the series
aoH-«i«oH + «,*oH
be less than a fixed number. Whenever this condition is ilHlil e d,
the series (2) converges absolutely for any value of x whose aboolnte
value is less than |xo|.
The number R is connected in a very simple wmy with the nnmbar w dsteed
in § 160, which is the greatest limit of the eeqaenoe
For if we coosider the analogous sequence
AiX, VATTi <a7X», .A^X-
378 SPECIAL SERIES [IX, §178
it Lb evident that the greatest limit of the terms of the new sequence is wX. The
sequence (1) therefore converges if X < 1/w, and diverges if X > 1/w ; hence
« = !/•».•
178. Continuity of a power seriea. Let f(x) be the sum of a power
series which converges in the interval from — R to -{- R,
(3) fix) = ao + «!« + • • • + a„«'» + • • • ,
and let /?' be a positive number less than R. We shall first show
that the series (3) converges uniformly in the interval from — R'
to -I- R'. For, if the absolute value of x is less than R', the
remainder R^
of the series (3) is less in absolute value than the remainder
of the corresponding series (1). But the series (1) converges for
A' = R', since R' < R. Consequently a number N may be found
such that the latter remainder will be less than any preassigned
positive number c whenever n'tN. Hence | i2„ | < c whenever n^N
provided that \x\<R\
It follows that the sum f(x) of the given series is a continuous
function of x for all values of x between — R and + R. For, let Xq
be any number whose absolute value is less than R. It is. evident
that a number R' may be found which is less than R and greater
than \xq\. Then the series converges uniformly in the interval
(— R', + R'), as we have just seen, and hence the sum/(cc) of the
series is continuous for the value Xq, since Xq belongs to the interval
in question.
This proof does not apply to the end points + R and — R of the
interval of convergence. The function f(x) remains continuous,
however, provided that the series converges for those values.
Indeed, Abel showed that if the series (3) converges for x = R, its
turn for z = Risthe limit which the sumf(x) of the series approaches
OM X approaches R through values less than R.\
L«t S be the sum of the convergent series
5 = a„ 4- «! /i 4- aai?^ -I- • • • + a^R'* H ,
• Thii theorem waa proved by Cauchy in his Coura d 'Analyse. It was rediscovered
by Hadamard in hie thesis.
t As stated above, these theorems can be immediately generalized to the case of
series of imaglQary terms. In tliiH case, liowover, care is necessary in formulating
the aenemllzaUon. See Vol. 11, § 2tJ«. — Trans.
IX. §17H] POWER SKRIE8 879
and let n be a positive mteger such that any one of the tiff
is less than a p reassigned positive number c If we set x « R$, and
then let $ increase from to 1, a; will increase from to A, and we
shall have
f(x) =f{eR) = Oo + aid/2 + a,^/?« + • •• + «.^i?- 4- • • •.
If n be chosen as above, we may write
(S -f(x) = a^R(l-e) + a, /?«(!- ^ + ... + a./?-(l - #^
(4) I 4- «,+ii2-*» 4- ••• -h a.^,/f-^' 4- • ••
i - a. + i^"**^""* a.+^^-^'Te--^' ,
and the absolute value of the sum of the series in the second line can-
not exceed e. On the other hand, the numbers ^**, ^"^ *,•••, ^♦^
form a decreasing sequence. Hence, by Abel^s lemma proved in f 75,
we shall have
In-t-l
'n+1
+ i/>n+i + . . . + a.^.^d--^"/?-*"! <0^*U< C.
It follows that the absolute value of the sum of the series in the
third line cannot exceed c. Finally, the first line of the right-hand
side of the equation (4) is a polynomial of degree t» in which
vanishes when ^ = 1. Therefore another positive number iy may be
found such that the absolute value of this polynomial is less than <
whenever 6 lies between 1 — t; and unity. Hence for all such values
of we shall have
|s-/i;x)|<3,.
But € is an arbitrarily preassigned positive number. Hence J{x)
approaches .S as its limit as x approaches /?.
In a similar manner it may be shown that if the series (3) eon-
verges for a; = — 72, the sum of the series for x = — 7? is equal to
the limit which /(a;) approaches as x approaches — R through valoei
greater than — R. Indeed, if we replace x by — x, this case redoeei
to the preceding.
' An application. This theorem enables OS to oomplsle the rtmlts of f 168
regarding the luuliiplication of series. Let
(5) S =Uo4i«i + m + . •. + «• + •••.
(6) 5' = i»o4»i4»t4--+».+ --
be two convergent series, neither of which convergM absolutely. The series
(7) WoOo + (uoti -4- U,t>o) + • • • + (uo». + • • • + «•••) + • • •
880 SPECIAL SERIES [IX, §179
in«y converge or diverge. If It converges, its sum S is equal to the product of
the sums of the two given series, i.e. S = SS\ For, let us consider the three
power series
/(X) = Mo + ttiX + . . • + U,X» + • • •,
^(x) = ro + ti« + •• + »««" + ••• »
f (Z) = UoVo + (Mot^i + UiVo)X + • • • + (Mot>» +••• + M„Vo) «" + ••• .
Each of these series converges, by hypothesis, when x = 1. Hence each of them
converges absolutely for any value of x between — 1 and + 1. For any such
Talue of X Cauchy*s theorem regarding the multiplication of series applies and
gives us the equation
(8) /(x)0(x) = ^(x).
By AbePs theorem, as x approaches unity the three functions /(x), ^(x), ^(x)
approach S, S\ and 2, respectively. Since the two sides of the equation (8)
meanwhile remain equal, we shall have, in the limit, Z = SS\
The theorem remains true for series whose terms are imaginary, and the proof
follows precisely the same lines.
179. Successive derivatives of a power series. If a power series
f(x) = ^0 + <^i^ 4- cLi^^ H h a„a^" H
which converges in the interval (— Rj -\- R) he differentiated term
by term, the resulting power series
•(9) ai-\-2a2X-\ + 7ia„a;"-^^H
converges in the same interval. In order to prove this, it will be
sufficient to show that the series of absolute values of the terms of
the new series,
^1 + 2^2^ + • • • + nA„X-' + . . .,
where i4, = |a^| and X = \x\, converges for X<R and diverges for
X>R.
For the first part let us suppose that X<R, and let /?' be a num-
ber between Xmd R, X<R'<R. Then the auxiliary series
R'^ R' R'^ R'Kr'J ^""^R'Kr'J ^"'
converges, for the ratio of any term to the preceding approaches
X/R't which is less than unity. Multiplying the successive terms
of this series, respectively, by the factors
IX, $179] POWER SERIES SS|
each of which is less than a certaia fixed number, since H'<H,w9
obtain a new series
which also evidently conTerges.
The proof of the second part is similar to the above. If the seriee
where A'j is greater than if, were convergent, the series
AiXi-\- 2AtX\ 4- • • • + «i4. AJ 4- • • •
would converge also, and consequently the series lA^X^ would con-
verge, since each of its terms is less than the corre8|>onding term of
the preceding series. Then R would not be the upper limit of the
values of A' for which the series (1) converges.
The sum /, (x) of the series (9) is therefore a continuous function
of the variable x inside the same interval. Since this series con-
verges uniformly in any interval (— 72', + /?'), where R*< R,fi(x)
is the derivative of f(x) throughout such an interval, by f 174.
Since R' may be chosen as near i2 as we please, we may aaeert that
the function /(a;) possesses a derivative for any value of x between
— R and + Ry and that that derivative is represented by the teriet
obtained by differentiating the given series term by term : •
(10) f{x) = ax 4- 2a^x -f ... 4- na^x!''' 4- •••.
Repeating the above reasoning for the series (10), we see thMi/{x)
has a second derivative,
/"(x) = 2a, 4- 6a,x 4. . . . 4- n(n - l)o.jf-» 4- •• •,
and so forth. The function /{x) possesses an unlimited sequence of
derivatives for any value of x inside the interval (— /?, 4- ^)» *n<i
these derivatives are represented by the series obtained by differen-
tiating the given series successively term by term :
(11) f<^\x) = 1 . 2 . . . na, 4- 2 . 3 . . . n(ii 4- 1)«, ♦ I ar 4- • .
If we set jr = in these formula, we find
ao=/(0), ai^AO), «. = ^'
or, in general,
1.2
• Although the oorreepondlng theorem to true for eeriee of li _
proof follows somewhat different linee. See VoL H, 1 3«. — TaAwe.
382 SPECIAL SERIES [IX, §179
The development oif(x) thus obtained is identical with the develop-
ment g^ven by Maclaurin's formula :
A')=AO) + ifXO) + ~r(0) + --- + ^-^/<'XO) + ....
The coefficients a^, a^ •••, a^y ••• are equal, except for certain
numerical factors, to the values of the function f(x) and its succes-
sive derivatives for x = 0. It follows that no function can have two
distinct developments in power series.
Similarly, if a power series be integrated term by term, a new
power series is obtained which has an arbitrary constant term and
which converges in the same interval as the given series, the given
series being the derivative of the new series. If we integrate again,
we obtain a third series whose first two terms are arbitrary ; and so
forth.
Examples. 1) The geometrical progression
1-x + x^-x* + "■ + (- 1)" a;" + • • •,
whose ratio is — x, converges for every value of x between — 1 and
4- 1, and its sum is 1/(1 + x). Integrating it term by term between
the limits and x, where |5c| < 1, we obtain again the development
of log(l -I- x) found in § 49 :
log(l + =») = MV I'-... + (-!)» gi4-....
This formula holds also for a; = 1, for the series on the right con-
verges when 05 = 1.
2) For any value of x between — 1 and + 1 we may write
Integrating this series term by term between the limits and x,
where |x|<l, we find
X X* a;* ^n + i
Since the new series converges for aj = 1, it follows that
f=i-|^g-7+-+(-i)"2-;n:i+-
IX. $m] POWKU SEHiES S8t
3) Let F(x) be the sum uf the convergent aeriee
F(x)^l-hjx+ ^^^ / j^^... + ^ 1.2- ./> ^ •^'>" •>
where m is any number whatever and |x| < 1. Then we shmll have
r(x) = m|_l + -p-x4.- + ^ li...\p-^) ' -'-'-^ -J-
Let lis multiply ea(;h side by (1 4- x) and then collect me lerms m
like powers of x. Using the identity
1.2 •.(/>-l) 1.2 ../» 1.2. />
which is easily verified, we find the formula
(1 + x)F'(*) = m [l + ^ , + !2i^i-|l) ^ + . . .
r»(m-l)...(m-p + l) 1
+ 1.2. -p '^* J
or
From this result we find, successively,
F'(x) ^ m
F(x) "l + a:*
log [F(x)] = m log (1 + x) + log r,
or
F(x)=C(l + x)-
To determine the constant C we need merely notice that F(P)^l.
Hence C = 1. This gives the development of (1 + x)* found in § fiO :
^ . m , , m(m — 1) • • • (m — » + 1) . ,
(l4-ar)'»=H- jx-f--- + -^ j 2...P '*' + •••.
4) Replacmg x by - x* and m by - 1/2 in the last formula abovt^
we find
y/lZT^^'^ ^ ^'^ 2.4.6... 2ii
This formula holds for any value of x between - 1 and -f 1 Inte-
grating both sides between the limits and «, whare we
obtain the following development for the arcsine
arc sin
x.lx» 1.3x»^ . 1.8.5 .(2ii-l; ^ ' ,
inx = j4-23+2746"^ *"*■ 2.4.6..2ii 2ii+l^ '
884
SPECIAL SERIES
[IX, § 180
180. Extension of Taylor's series. Let /(a:) be the sum of a power
series which converges in the interval (— R, + R), x^ a point inside
that interval, and x^ + h another point of the same interval such
that |av,| +1 A| < ii. The series whose sum is /{x^ -}- A),
«o + ai(«o 4- A) + <h{x^ + A)' + • • • + a„(«o + hY + "'y
may be replaced by the double series obtained by developing each
of the powers of {xq + h) and writing the terms in the same power
of h upon the same line :
(12)
This double series converges absolutely. For if each of its terms
be replaced by its absolute value, a new double series of positive
terms is obtained:
ao + «i«o4- ajicj +•
•+ a.x^o +•••
-\-aih -h2a^Xoh-{'-
• + n a^x^-^h -\
+ a,A« +•
• + ^^^«X-'A^ + -
+ •
(13)
n A,\x^\-^\h\ +.
H-
If we add the elements in any one column, we obtain a series
which converges, since we have supposed that | iCo | + 1 A | < /?. Hence
the array (12) may be summed by rows or by columns. Taking
the sums of the columns, we obtain f{xQ-\-h). Taking the sums
of the rows, the resulting series is arranged according to powers of
A, and the coefficients of h, A^^ ... are f\x^),f%x^)/2\,'--, respec-
tively. Hence we may write
(14) f(x. + h) =/(Xo) -f \f\x,) + . . . + j-ilL_^n)(^^) + . . .^
If we assume that |A|<i2-|xJ.
This formula surely holds inside the interval from x^ - R -[■\xq\
to a!o + ^-|a?o|» but it may happen that the series on the right
converges in a larger interval. As an example consider the function
IX. § 180] POWER SERIES $^
(l+ar)"*, where m U not a positiye integtr. The dtfrelopoMiil
according to powers of x holda for all values of x between ~ 1 and
+ 1. Let Xq be a value of x which lies in that interval. Than we
may write
(1 + «)-= (1 4- a^ 4- X - x,)- = (1 + «,)-(! + «)-,
where
« = -2.
We may now develop (1 -f «)"* according to powers of s, and this
new development will hold whenever |«| < 1, Le. for all valaes of x
between — 1 and 1 + 2aro. If Xo is positive, the new interval will be
larger than the former interval (—1, -f 1). Hence the new formula
enables us to calculate the values of the function for values of the
variable which lie outside the original interval. Further inveetig»»
tion of this remark leads to an extremely important notion, — that
of analytic extension. We shall consider this Subject in the sefwod
volume.
Note. It is evident that the theorems proved for series arranged
according to positive powers of a variable x may be extended
ately to series arranged according to positive powers of x — «,
more generally still, to series arranged according to poeitive
of any continuous function ^(x) whatever. We need only consider
them as composite functions, ^(x) being the auxiliary fnnctioiL
Thus a series arranged according to positive powers of l/x eon-
verges foT all values of x which exceed a certain positive constant in
absolute value, and it represents a continu ous fun ction of x for all
such values of the variable. The function Vx* — a, for example, may
be written in the form ± x(l — a/x^*. The expression (1 — e/*^*
may be developed according to powers of l/x* for all values of •
which exceed Va in absolute value. This gives the formula
Vx a-x 2" 2.4a,. • • 2.A,6 2p x»'
which constitutes a valid development of Vx* — a whenever x > v«.
When X < — V a, the same series converges and represents the func-
tion — Vx* - a. This formula may be used advantageously to obCaio
a development for the square root of an integer whenever the fifft
perfect sijuare which exceeds that integer it known.
386 SPECIAL SERIES [IX, § 181
181. Dominant functions. The theorems proved above establish a
close analogy between polynomials and power series. Let (—r, -\-r)
be the least of the intervals of convergence of several given power
series /i (j*), /a (x), •••,/„ (a:). When |a:|<r, each of these series
converges absolutely, and they may be added or multiplied together
by the ordinary rules for polynomials. In general, any integral poly-
nomial in fi (x)y fi (x), • • • , /n (x) may be developed in a convergent
power series in the same interval.
For purposes of generalization we shall now define certain expres-
sions which will be useful in what follows. Let f(x) be a power
series
f(x) = tto + dix + a^x^ H h a„a;" ■] ,
and let ^(x) be another power series with positive coefficients
^(a;) = n'o + «ix + oTgX^H h «'„x" H
which converges in a suitable interval. Then the function <f>(x') is
said to dominate * the function f{x) if each of the coefficients a„ is
greater than the absolute value of the corresponding coefficient of
|ao|<«'oj |ai|<«ij •••, |a«|<ar„, •••.
Poincar^ has proposed the notation
f{x)<4>(x)
to express the relation which exists between the two functions f{x)
and <^(x).
The utility of these dominant functions is based upon the fol-
lowing fact, which is an immediate consequence of the definition.
Let P(aQy «!, •••, a„) be a polynomial in the first w +1 coefficients
of f{x) whose coefficients are all real and positive. If the quanti-
ties ao» *i» •••> o,n be replaced by the corresponding coefficients of
^x), it is clear that we shall have
For instance, if the function <j>{x) dominates the function /(x),
the series which represents \_<^{x)Y will dominate [/(x)]^ and so
on. In general, i<i>{x)Y will dominate [/(x)]«. Similarly, if <^ and
^1 are dominant functions for / and /^ , respectively, the product </)<^i
will dominate the product ^i; and so forth.
•Thl« expremion will be used as a translation of the phrase " 4>{z) est majorante
poor la fomiIon/(z)." Likewise, "dominant functions " will be used for " fonctions
inajorauuoM." — Trams.
IX,} 181] POWER 8KRIK8 $87
Given a power seriee/ljx) which converges in an intenral (— J?, + JT),
the problem of determining a dominant function is of ooane iodeter-
minate. Hut it is convenient in what follows to make the domi-
nant function as simple as possible. Let r be any number lass *hv n
R and arbitrarily near R. Since the given series converges for jr « r,
the absolute value of its terms will have an upper limit which w«
shall call M, Then we may write, for any value of f»,
A,f*<M or |«.| = ^.<~
Hence the series
r
whose general term is M(x*/t*)f dominates the given function /l[x).
This is the dominant function most frequently used. If the series
f(x) contains no constant term, the function
r
may be taken as a dominant function.
It is evident that r may be assigned any value less than R, and
that M decreases, in general, with r. But Af can never be less than
Aq. If .lo is not zero, a number p less than R can always be found
such that the function i4o/(l — x/p) dominates the function /^x).
For, let the series
3/ -f- . V - + .1/ ^ + . . • -f- 3/ ^ 4- • • • ,
where M > A^^ be a first dominant function. If p be a number less
than vAq/M and n > 1, we shall have
,p.| = |a.^|x(e)"<«?(?)"
whence |a^p"| < vIq. On the other hand, |a,|=3^,. Hence the
X X^ jpB
Ao + A,--^ A.j^-h •" + A.-A-
dominates the function f(x). We shall make use of this fact pree»
ently. More generally still, any number whatever which is gieater
than or equal to A^ may be used in plaoe of M.
888 SPECIAL SERIES [IX, §182
It may be shown in a similar manner that if Aq = 0^ the function
18 a
dominant function, where /i is any positive number whatever.
Note, The knowledge of a geometrical progression which dominates the fanc-
tion /(x) also enables us to estimate the error made in replacing the function
/{z) by the sum of the first n + 1 terms of the series. If the series M/{1 — x/r)
dominates /(x), it is evident that the remainder
of the given series is less in absolute value than the corresponding remainder
of the dominant series. It follows that the error in question will be less than
,(•)■
M
1-5
182. Substitution of one series in another. Let
(16) z =f(y) = tto + aiy + • • • + a„2/" + • • •
be a series arranged according to powers of a variable y which con-
verges whenever \y\<R. Again let
(16) y = <f>(x) = b^-\-b,x + ...^ b^x^ + . . .
be another series, which converges in the interval (— r, + r). If
y» y\ y*, • • • in the series (15) be replaced by their developments in
series arranged according to powers of x from (16), a double series
<h + axho + a^bl +•••+ aj-. +...
(17) . + «i*ia^ + 2aa b^b^x + • • • + na^b^-^b^x -\
4- a^b^x^ + a^(b\ -h 2b^b^)x^ -f
is obtained. We shall now investigate the conditions under which
this double series converges absolutely. In the first place, it is
necessary that the series written in the first row,
IX, § 182] POWER SERIES SS9
should converge abtolntely , i.e. that | b^ \ should be leM than R.^ This
eondition is also mffieient. For if it is satisfied, the funotioD 4(x)
will be dominated by an expression of the form m/{l — x/p)^ whaie
m is any positive number greater than |6,| and where p<r. We
may therefore suppose that m is less than R. Let R' be aaolber
positive number which lies between m and R. Then the funetioo
f(y) is dominated by an expression of the form
R'
If y be replaced by m/(l — x/p) in this last series, and the powera
of 1/ be developed according to increasing powers of x by the binomial
theorem, a new double series
(18)
is obtained, each of whose cociiicuMiis r than
the absolute value of the corresponding vv (17),
since each of the coefficients in (17) is formed from the coeflioienta
<^oy <'^if f^if " f Ki hi ^if ' • ' ^y means of additions and multiplicattoos
only. The double series (17) therefore converges abeolutelj pro-
vided the double series (18) converges absolutely. If x be replaoed
by its absolute value in the series (18), a necessary condition for abeo-
lute convergence is that each of the series formed of the terms in any
one column should converge, i.e. that |x| < ^ If this oonditioQ be
satisfied, the sum of the terms in the (n + l)th column is equal to
M
_«■(!- f)J
Then a further necessary condition is that we shoald luive
(19) \'^<f'{^'-l)'
or
• The case in which the series (15) conrerget for9»ir(M«fl77)wmh»
In what follows. — Trans.
390 SPECIAL SERIES [IX, § 182
Since this latter condition includes the former, |a;| <p, it follows
that it is a necessary and sufficient condition for the absolute con-
vergence of the double series (18). The double series (17) will
therefore converge absolutely for values of x which satisfy the
inequality (19). It is to be noticed that the series </>(x) converges
for all these values of x, and that the corresponding value of y is
less than 72' in absolute value. For the inequalities
9
necessitate the inequality |<^(a;)| < iJ'. Taking the sum of the series
(17) by columns, we find
ao 4- a^<^{x) + a^{^^{x)J + • • • + a„[<^(«)]'' + • • •,
that i8,/[<^(J7)]. On the other hand, adding by rows, we obtain a
series arranged according to powers of x. Hence we may write
(20) /[<^(a;)] = c, + c^x + c^x?- + . . . -f c^x- + • • .,
where the coefficients c^, Cj , Cg, • • • are given by the formulae
(21)
f Co = ao + ai^»o + tta^ -\ f- aj)l -\ ,
which are easily verified.
The formula (20) has been established only for values of x which
satisfy the inequality (19), but the latter merely gives an under
limit of the size of the interval in which the formula holds. It may
be valid in a much larger interval. This raises a question whose
solution requires a knowledge of functions of a complex variable.
We shall return to it later.
Special cases. 1) Since the number W which occurs in (19) may
be taken as near R as we please, the formula (20) holds whenever x
satisfies the inequality \x\<p(l- m/R). Hence, if the series (15)
converges for any value of y whatever, R may be thought of as infinite,
p may be taken as near r as we please, and the formula (20) applies
whenever |x| < r, that is, in the same interval in which the series
(16) converges. In particular, if the series (16) converges for all
values of x, and (16) converges for all values of y, the formula (20)
is valid for all values of x.
IX. § 182] POWER SERIES 891
2) When the constant term b^ of the series (16) is nra^ tbm fun^
tion <f>(x) is dominated by an expression of the form
m
m,
■-I
where p <r and where m is any positive number whatever. An
arguinent similar to that used in the general case shows that the
formula (20) holds in this case whenever x satisfies the inequality
(22) I'KpFT^'
where /^' is as near to /? as we please. The corresponding inierral
of validity is larger than that given by the inequality (19).
This special case often arises in practice. The inequality
|/>o| < /2 is evidently satisfied, and the coefficients <r. depend upon
«oj «i> •••> «»> ^n ••> ^» only:
Co = «o> Ci = aibiy Ca = ttift, -h a,^, = «t*« "I + «.*?•
Examples. 1) Cauchy gave a method for obtainhig the binomial theorem from
the development of log(l + x). Setting
y = ;xlog(l + x) = M(^j--+3-^-.-j.
we may write
(1 + x)** = eMioeO + ') = c» = l + ^+|^ + ---,
whence, substituting the first expansion in the Mcond,
/x x« x« \ u« /x «> x« \«
If the right-hand side be arranged according to powers of c. It is ertdeni thai
the coefficient of x" will be a polynomial of degree a in m whieh we ahaU eall
P^(m). This polynomial must vanish when m = 0, 1» 2, • • •, a - 1, end moiS
reduce to unity when fi = n. These facta completely detarmiBe P« la the foca
<^^) ^"^ 1.2...»
2) Setting « = (1 + x)'/', where x lies between - 1 and -i- 1, we may writ*
whore
» = l.og(l + .)-l-| + f — • + (-1)--'^-
892 SPECIAL SERIES [IX, §183
The firet expansion is valid for all values of y, and the second is valid whenever
|x|< 1. Hence the formula obtained by substituting the second expansion in
the first holds for any value of x between - 1 and -f 1. The first two terms of
this formuU are
,^)(l + x)i = .-|(l + l + ji^ + ... + :^^ + ...) + -- = e-?x + ....
It follows that (1 + x)i/' approaches e through values less than e as x approaches
Mro through positive values.
183. Division of power series. Let us first consider the reciprocal
of a power series which begins with unity and which converges in
the interval (— r, -h ?')• Setting
y = bix -{- b^x^ -\ ,
we may write
whence, substituting the first development in the second, we obtain
an expansion for f(x) in power series,
(25) f(x)=^l-b,x-^(bl-b,)x^ + .--,
which holds inside a certain interval. In a similar manner a devel-
opment may be obtained for the reciprocal of any power series
whose constant term is different from zero.
Let us now try to develop the quotient of two convergent power
series
4*(^) _ gp + <^i^ + <^2^^ H
Xl/(X) bo-\-biX+ b2X^-\----'
If 6o is not zero, this quotient may be written in the form
Then by the case just treated the left-hand side of this equation is the
product of two convergent power series. Hence it may be written
in the form of a power series which converges near the origin ;
Clearing of fractions and equating the coefficients of like powers
of 3P, we find the formulu3
DC. § \m] POWER 8£KI£8 S9S
(27) a, = b,e, + *»«._, + . . . + i.e., !• - 0, 1, IV
from which the coefficients ^» 0|» "•) 0. mij be wJ m Ut^
sively. It will be noticed that these ooefRoieoti Kn the
those we should obtain by performing the diyision indioated by the
ordinary rule for the division of polynomials arranged aoooidiiig to
increasing powers of x.
If ^0 = ^> th® result is different. Let us suppose for generality
that \f/(or) = jr*^\J/i(x), where A; is a positive integer and ^,(jt) is a
])()wei* series whose constant term is not zero. Then we may write
»(x) ^ 1 (^r,>
and by the above we shall have also
-p^ = Co + Ci* 4- • • • 4- e.^ix"'' + c»2J* + «t^,«**» 4- • • .
It follows that the given quotient is expressible in the form
(28) ii^ = $. + -£u + ... + £^. + ,.+ - -, ,
^ ^ \f/(x) aj* a** ' X
where the right-hand side is the sum of a ratio: .vhich
becomes infinite f or x = and a power series whu .. ...... v. 5^^ near
the origin.
Note. In order to calculate the successive powers of a power Mrits, it is
venient to proceed as follows. Assuming the identity
(oo + aix + • • • + a^.x" + .••)"• = Co + Cii + ••• + c,x« + • • -,
let us take the logarithmic derivative of each side and than d^r of frmrtions.
This leads to the new identity
12Q\ (»»(ai + 2a«x+ • • + na,x--» + • • .)(eo + CiX + • • + c,x« + ♦• •)
^^' j = (ao + aix + . • + OnX* + • • '){ci + 2c«* + • • + «c.x— » + .••).
The coefficients of the various powers of x are easily ealcalaied. Squat-
ing coefficients of like powers, we find a aeqaMiM of fornmUs froai whidi
Co, ci, . . ., c, . • • may be found saoce«lvdy if Co be known. It is evIdMl that
184. Development of l/Vl - Sxi + f«. Let us dtfelop l/VI - iM + f«
according to powers of x. Setting y = 2xt - i«, we shaU have, when Iff < 1.
y)-*=1 + ly+l^|f-^ .
or
(30) -_i=_ = i + ??!fi' + ?(l«-.V + -.
1 _.,
1
vr
-2x« + i«
894 SPECIAL SERIES [IX, §185
Collecting the terms which are divisible by the same power of «, we obtain an
expansion of the form
(81) ■ ^ = Po + Piz + Ptz^ + •" + P»«" + • • •,
where
Po = l, Pi = «, Pa = -^^»
and where, in general, P» is a polynomial of the nth degree in x. These poly-
nomials may be determined successively by means of a recurrent formula. Dif-
ferentiating the equation (31) with respect to z, we find
= Pi + 2P82 + • . . + nPnZ'*-! + • • •,
(1 - 2X2 + 3fl)*
or, by the equation (31),
(X - 2)(Po + PiZ + • • • + P„2" + • • •) = (1 - 2x« + z^){Pi + 2P2Z + ").
Equating the coefficients of z", we obtain the desired recurrent formula
(n + l)P« + i = (2n + l)xP„ - nPn-i .
This equation is identical with the relation between three consecutive Legendre
polynomials (§ 88), and moreover Po = Xq, Pi = Xi, P2 = X2 . Hence P„ = X„
for all values of n, and the formula (31) may be written
(82) , ^ = 1 + Xi 2; + X2 22 + . . . + x„ 2» + . . . ,
Vl - 2X2 + 2*
where X, is the Legendre polynomial of the nth order
X- = 5 — r(x2 - IH .
2.4.6...2n tte«'-^ '■■
We shall find later the interval in which this formula holds.
II. POWER SERIES IN SEVERAL VARIABLES
185. General principles. The properties of power series of a single
variable may be extended easily to power series in several independ-
ent variables. Let us first consider a double series ^a„^x'"y'', where
the integers m and n vary from zero to + 00 and where the coeffi-
cients a^, may have either sign. If no element of this series exceeds
a certain positive constant in ahsolute value for a set of values
* = *o» y = yo> ^^« series converges absolutely for all values ofx and
y which satisfy the inequalities |«| < |«o|, |y| < |2/o|-
For, suppose that the inequality
I««.a7y5|<3f or \a^,\< ^
l^oriyol"
IX, $180] DOUBLE POWER SERIES |M
is satisfied for all sets of values of m and fi. TbaD Uis abioloie valiM
of the general element of the double series Sa.«z*y* is less than the
corresponding element of the double series 1M\x/x^^\yfyJ^, Bat
the latter series converges whenever |'|<|2^|> |y|<|yt|i and its
sum is
M
('-i^i)('-i^l)
as we see by taking the sums of the elements by oolomns and
adding these sums.
Let r and p be two positive numbers for which the double
^la^nl'^p'* converges, and let R denote the rectangle formed by the
four straight lines x = r, x = — Vf y = p, y =— p. For every point
inside this rectangle or upon one of its sides no element of the
double series
(33) F(x,y) = 2a..a-y
exceeds the corresponding element of the series Sjanil'^P* ^ abso-
lute value. Hence the series (33) converges absolutely and uni-
formly inside of R, and it therefore defines a continuous function
of the two variables x and y inside that region.
It may be shown, as for series in a single variable, ll»at the
double series obtained by any number of term-by -term differen-
tiations converges absolutely and uniformly inside the rectangle
bounded by the lines a* = r — c, x = — r + €,y = p — t*, y»— ^ + €',
where e and e' are any positive numbers less than r and ^ respec-
tively. These series represent the various partial derivatives of
F{x, y). For example, the sum of the series SfiMi.,«"-V is equal
to cF/dx. For if the elements of the two series be arranged aeeofd*
ing to increasing powers of a?, each element of the second series it
equal to the derivative of the corresponding element of the ilrtt
Likewise, the partial derivative d*-^'F/djrd^ is equal to th<» sum
of a double series whose constant factor is a.,1.2-»' ^.
Hence the coefficients e?^. are equal to the vmlues of the corrMpond-
ing derivatives of the function F{x^ y) at the point x « y k 0, except
for certain numerical factors, and the formula (SS) may be written
in the form
896 SPECIAL SERIES [IX, §186
It follows, incidentally, that no function of two variables can have
two distinct developments in power series.
If the elements of the double series be collected according to
their degrees in x and y, a simple series is obtained :
(36) F(a;, y) = «^o + <^i + <A2 + • • • + ^« + • • • »
where ^, is a homogeneous polynomial of the wth degree in x and
y which may be written, symbolically,
1 / dF dFV'O
The preceding development therefore coincides with that given by
Taylor's series (§ 51).
Let (a^, yo) ^ a Poi°* inside the rectangle R, and (xq + A, yo 4- k)
be a neighboring point such that | iCo | + 1 ^ | < r, | yo | + 1 A; | < p. Then
for any point inside the rectangle formed by the lines
x = Xo±[r-\Xo\^y 3^ = yo±[p-|2/o|],
the function F(Xf y) may be developed in a power series arranged
according to positive powers of x — Xq and y — y^i
(36) n^. + .,yo + ^) = Z ,;Sg..n ^'"^-
For if each element of the double series
be replaced by its development in powers of h and k, the new multi-
ple series will converge absolutely under the hypotheses. Arrang-
ing the elements of this new series according to powers of h and k,
we obtain the formula (36).
The reader will be able to show without difficulty that all the
preceding arguments and theorems hold without essential altera-
tion for power series in any number of variables whatever.
186. Dominant functions. Given a power series f(x, y^ z, -- •) in n
yariables, we shall say that another series in n variables <^(ic, y,z,- •)
damiruUes the first series if each coefficient of </>(a;, y, «, • • •) is positive
and greater than the absolute value of the corresponding coefficient
^^ A*» Vt "t " •)• The argument in § 185 depends essentially upon
IX. § iHB] DOUBLE POWER SSaiES SOT
the use of a dominant function. For if the feriof S|«_,jr»y»| eoD*
verges for x = r, y = p, the function
where M is greater than any coefficient in the aeriee Sla^.r^W^L
dominates the series 2a^,a!"y". The function
^{xy y) =
is another dominant function. For the coefficient of jr*y' in \,{x, >/,
is equal to the coefficient of the corresponding term in thr rxnan
sion of Af(x/r -f y/p)'*-^*, and therefore it is at least equal to the
coefficient of x^y* in <f>(x, y).
Similarly, a triple series
f{x, y, x) = Sa„^ary .I*,
which converges absolutely for x ^ r^ y ^r'f M^r'*^ whei« r, r*, r"
are three positive numbers, is dominated by an expreation of the
form
^{Xy yyz) =
('-f)('-5)('-^)
and also by any one of the expressions
lff{x, y, z) contains no constant term, any one of the pr «<<*«itTy f Hp *ff
sions diminished by M may be selected as a dominant fonetioo.
The theorem regarding the substitution of one power sec i ei In
another (§ 182) may be extended to power seriee in seTeral rariahlaa.
If each of the variables in a amver^ertt pomer $miu in p v mriM m
yiy ytf "t yp^^ replaced by a eo mver^m i jwwer ssrtts m f gart aU si
xi, Xiy "jX^ which hcu no eomiant fsna, tks retuU tf Ms snAsTilii-
tion may be written in the form of a power <
to powers of x^, x^, ■, x^, provided iksU iks mU§t m t 4 mIm tf <
of these variables is Uss thttn a
898 SPECIAL SERIES [IX, §186
Since the proof of the theorem is essentially the same for any
number of variables, we shall restrict ourselves for definiteness to
the following particular case. Let
(37) i^(y,«)=2a„„2r«"
be a power series which converges whenever | y | ^ r and | « | < r', and let
^ ' \ z = c^x -\- c^x^ -\ + c„a;«H
be two series without constant terms both of which converge if the
absolute value of x does not exceed p. If y and z in the series (37)
be replaced by their developments from (38), the term in ?/'"«'• becomes
a new power series in x, and the double series (37) becomes a triple
series, each of whose coefficients may be calculated from the coeffi-
cients a„„, h^f and c„ by means of additions and multiplications
only. It remains to be shown that this triple series converges abso-
lutely when the absolute value of x does not exceed a certain con-
stant, from which it would then follow that the series could be
arranged according to increasing powers of x. In the first place,
the function /(y, z) is dominated by the function
and both of the series (38) are dominated by an expression of the form
(40) JL.-N=f^N($j,
1--
where M and N are two positive numbers. If y and z in the double
series (39) be replaced by the function (40) and each of the products
y**" be developed in powers of cc, each of the coefficients of the result-
ing triple series will be positive and greater than the absolute value
of the corresponding coefficient in the triple series found above. It
will therefore be sufficient to show that this new triple series con-
verges for sufficiently small positive values of x. Now the sum of
the terms which arise from the expansion of any term ?/'»«'• of the
•eries (39) is
/x\^ + "
■^" VpI
" (i-r"'
IX, J 187] REAL ANALYTIC FfVrTTOVS 999
which is the general term of the seriea outamed by XDuitiplviog
two series
i(?)-Ui
term by term, except for the constant factor J/. Both of (be
series converge if x satisfies both of the inequalitiM
^ < pz-rn}' ' < P
It follows that all the series considered will conrerge abtolatelj,
and therefore that the original triple series may be arranged aooord-
iug to positive powers of x^ whenever the absolute value of x U lets
than the smaller of the two numbers pr/{r + N) and ^/{f + N),
Note. The theorem remains valid when the series (38) oontmin
constant terms b^ and Cq^ provided that |6o| "^^ ^ <u>d |<^|< r'. For
the expansion (37) may be replaced by a series arranged according
to powers oi y — bo and 2; — Co> by S 185, which reduces the disoos-
sion to the case just treated.
m. IMPLICIT FUNCTIONS
ANALYTIC CURVES AND SURFACES
187. Implicit functions of a single variable. The existence of implieil
functions has already been established (Chapter II, { 20et If.) luidw
certain conditions regarding continuity. When the left-hand sides
of the given equations are power series, more thorough investigmtioii
is possible, as we shall proceed to show.
Let F(Xy 1/)= be an equation wfune ^ft-kamd sids can be <
in a convergent power series arranged aoeordmg to u
of X — Xo and y — yo> v)here the eenstani term is sem
cient of 1/ - y^f is different from Mere, Thm the equation ha$ erne mmd
only one root which approaehee y^ae x approaekoe a^, emd tkmi rmi
can be dei^eloped in a power series arramged artordinp to powers of
X — x«.
For simplicity let us suppose that x^^y^^O, which aaMmnts lo
moving the origin of cottrdinates. Transposing the term of the first
degree in y, we may write the given equation in the form
(41) y =yi;x, y) = a,.z + a,oX* + <hi^ + ««/ + ' '.
400 SPECIAL SERIES [IX,§ia7
where the terms not written down are of degrees greater than the
second. We shall first show that this equation can be formally sat-
isfied bj replacing y by a series of the form
(42) y = ciaj H- Cax' + . . • 4- c^x^ + • • •
if the rules for operation on convergent series be applied to the series
on the right. For, making.the substitution and comparing the coeffi-
cients of Xj we find the equations
and, in general, c, can be expressed in terms of the preceding c's
and the coefficients a,^., where i + A; < n, by means of additions and
multiplications only. Thus we may write
(43) C, = PniP'lQl «20J ^\li ' -J «0»)>
where P, is a polynomial each of whose coefficients is a positive
integer. The validity of the operations performed will be estab-
lished if we can show that the series (42) determined in this way
converges for all sufficiently small values of x. We shall do this by
means of a device which is frequently used. Its conception is due
to Cauchy, and it is based essentially upon the idea of dominant
functions. Let
<^(x, y)=2i„„x'»y»
be a function which dominates the function /(a;, y), where b^^ = J^j =
and where b^^ is positive and at least equal to |a„,„|. Let us then
consider the auxiliary equation
(41') Y^^(^^^Y)=:S,b„„x^Y-
and try to find a solution of this equation of the form
(42^) Y=:C^x+ Caa;«-f ...-}-C,a;" + --«.
The values of the coefficients Ci, C,, • • • can be determined as above,
and are
C, = *,o, C, = b,o + bnC, + bo,Cly ' . .,
and in general
It is evident from a comparison of the formulas (43) and (43')
that |c,| < C,, since each of the coefficients of the polynomial P„ is
positive and |a.J<6^,. Hence the series (42) surely converges
DC, 5 187] REAL ANALYTIC FUNCTI0V8 401
whenever the series (42^) conyerges. Now we maj mImI tu th$
dominant fuii('fi«m ehi' r y\ tlie function
where 3f, r, and p are three positive numbers. Then the auxiiuu^
equation (41') becomes, after clearing of fraction*,
p + M p + M l__x^
r
This equation has a root which vanishes for 2 = 0, namelj:
2(p4-Af) 2(p4-3/)\/^ p« l^l*
The quantity under the radical may be written in the form
(■-3(-9-'
Hence the root }' may be written
It follows that this root Y may be developed in a eeries which eon-
verges in the interval (— a, -f <t), and this development must coin-
cide with that which we should obtain by direct substitution, that
is, with (42'). Accordingly the series (42) converges, a foriimrt^ in
the interval (— or, + a). This is, however, merely a lower limit of
the true interval of convcrutMU!** of tlie series <'42^. whir.h may be
very much larger.
It is evident from the mannrr ::
determined that the sum of the series Mj, s.i:j,i,. :: . , , .a_ v ,U.
Let us write the equation F(«, y) in the form y — /[«, y) « 0, and
let y — P(x) be the root just found. Then if P(x) -f s be tabeti-
tuted for i/ in F(7, y), and the result be arranged accordiDg lo
powers of x and z^ each term must be divisible by «, sinre the whole
expression vanishes when « = for any value of x. We aball ha^a
then Fix, P(x) -f «] = xQ(x, x) , where Q(x, «) is a power teriee in m
where
402 SPECIAL SERIES [IX, § 188
and z. Finally, if z be replaced by y — P{x) in Q(Xy z), we obtain
the identity
F{x,y) = ly-P(x)-]Q,{x,y),
where the constant term of Qx must be unity, since the coefficient
of y on the left-hand side is unity. Hence we may write
(44) F{x, y) = [y - P{x)-](1 + ax + ^y + • • •)•
This decomposition of F{xj y) into a product of two factors is due
to Weierstrass. It exhibits the root y = P(x), and also shows that
there is no other root of the equation F(xj y) = which vanishes
with x, since the second factor does not approach zero with x and y.
Note. The preceding method for determining the coefficients c„ is
essentially the same as that given in § 46. But it is now evident
that the series obtained by carrying on the process indefinitely is
convergent.
188. The general theorem. Let us now consider a system of p equa-
tions inp -\- q variables.
'Fi(a:i,X2, •..,«,; yi, y2, " •, yp)= 0,
^2(2:1, 0^2, •••,«,; yi,y2y"-,yp)=0,
(46)
Fp(xi,Xi,...,Xg] yi,2/2, •••,yp)=0,
where each of the functions F^, F^, •■, Fp vanishes when Xi = y^ = 0,
and is developable in power series near that point. We shall further
suppose that the Jacobian D(Fi, F^, •••, Fp)/D{y^, 2/2, ••, ^p) does
not vanish for the set of values considered. Under these conditions
there exists one and only one system of solutions of the equations (45)
of the form
Vi = <^i(«i, a^a, '-,Xq), . . ., y^ = <^^, (^i, Xj, . . ., a;,),
where <^i , <^a , • • , <^p are power series in x^, x^, • • ■, Xg which vanish
when a?! = X, = . • . = iB, = 0.
In order to simplify the notation, we shall restrict ourselves to
the case of two equations between two dependent variables u and v
and three independent variables x, y, and z :
(46) )^i=aw +bv -{-ex -\-dy -\- ez +... = 0,
F, = a'w -f Vv + c'x + c^'y 4. e'« ^ = 0.
Since the determinant ab' - ba' is not zero, by hypothesis, the two
equations (46) may be replaced by two equations of the form
IX, $188] REAL ANALYTIC FUNCTIONS 408
(47) (u-Ja^^x-y-.'ii'-',
where the left-hand sides contain no constant leruis and no tenu
of the first degree in u and t;. It is easy to show, as iboTe, tfaH
these equations may be satisfied formalLy by replacing m and v by
power series in Xy y^ and z :
(48) u = 2c,,,aryV, v = 2r;^,x'/«',
where the coefficients e^,^, and el^, may be calculated from a^M^ and
^mnpqr ^J mcans of additions and multiplications only. In order to
show that these series converge, we need merely compare them with
tlie analogous expansions obtained by solving the two auxiliary
equations
— (.■^^;(..^) -'(--^>
where M, r, and p are positive numbers whose meaning hat been
explained above. These two auxiliary equations reduce to a single
equation of the second degree
2p-f 43f^2p4-43/ ._x+x±f
r
which has a single root which vanishes for x = y
s M
£ e
^ 4(p + 23/) 4(p + 2Af)
where a = r [p/(p + 43/)]^
This root may be developed in a convergent power series when-
ever the absolute values of x, y, and « are all less than or equal to
a/3. Hence the series (48) converges under the same oooditioos.
Let wi and v^ be the solutions of (47) which are deralopable in
series. If we set u = u^ -f m', v = rj + v' in (47) and artmnge tht
result according to powers of x, y, «, «', v\ each of the terms mosl
be divisible by u' or by v'. Hence, returning to the original ▼aria-
bles Xj y, «, w, v, the given equations may be written in the form
a7'^ Ut*-tiO/+(t'-«'i)^ =0,
^^'^ J(u-ti,)/i4-(p-«'i>^i = 0,
404 SPECIAL SERIES [IX, §189
where /, ^, /i , ^j are power series in a;, y, «, w, and v. In this
form the solutions u=.ti^yV = Vi are exhibited. It is evident also
that no other solutions of (47') exist which vanish for a; = y = « = 0.
For any other set of solutions must cause /<^i — </>/i to vanish,
and a comparison of (47) with (47') shows that the constant term
is unity in both / and ^i, whereas the constant term is zero in
both/j and ^; hence the condition /<^i — <j>fx = cannot be met by
replacing u and v by functions which vanish when x = y = z=.0.
189. Lagntnge't formaU. Let us consider the equation
(40) y = a + a^(y),
where ^(y) is a function which is developable in a power series iny —a^
^(y) = 0(a) + (y - o) <t>\o) + ^^^f^ r{a) + . . . ,
which converges whenever y — a does not exceed a certain number. By the
general theorem of § 187, this equation has one and only one root which
approaches a as z approaches zero, and this root is represented for sufficiently
•mall values of x by a convergent power series
y = a + aiz + a^x^ -\ .
In general, if f{y) is a function which is developable according to positive
powers of y — a, an expansion of /(y) according to powers of x may be obtained
by replacing y by the development just found,
(50) f{y) =fia) + ^ix + ^2X2 + . . . + ^^a.» + : . . ^
and this expansion holds for all values of x between certain limits.
The purpose of Lagrange's formula is to determine the coefficients
■^Ij -0.2, • • •, Any ' • •
in t«rms of a. It will be noticed that this problem does not differ essentially
from the general problem. The coefficient A^ is equal to the nth derivative of
AV) 'or y = 0, except for a constant factor n!, where y is defined by (49); and
this derivative can be calculated by the usual rules. The calculation appears to
be very complicated, but it may be substantially shortened by applying the fol-
lowing remarks of Laplace (cf. Ex. 8, Chapter II). The partial derivatives of
the function y defined by (49), with respect to the variables x and a, are given
by the formuls
[l-x«'to)]^ = «(„), tl-X«'(i,)]^ = l,
we find Immediately
<"' k"=*<<"-
"f^^f K = /(y). On the other hand, It is easy to show that the formula
IX,J18»] REAL ANALYTIC FUNCTIONS 4%
b identically aatltfled, where F{y) is ad arbitimry fonelioa of y. f^
■ide become!
OD performing the indicated differentiations. We ahall now provw
^''^>[*^>'m
for any value of n. It holds, by (61), for n = 1. In order to profo It la §m^
eral, let us assume that it holds for a certain number n. Then we dMll iHift
But we also have, from (51) and (51'),
whence the preceding formula reduces to the form
dn^
which shows that the formula in question holds for all valaee of a.
Now if we set X = 0, y reduces to a, u to /(a), and the nth derlfttito oi «
with respect to 2 is given by the fonnula
/a*u\ _ d— »
\ax" /o ~ da"-»
[^a)-/'(a)],
Hence the development of /(y) by Taylor's series beooi
(52)
This is the noted formula due to LagraofB. It givet an 9t^ 9m\ fm for tlw
root y which approaches zero as x i4)proaohes aero. Wo iriiaU tad lattr tiM
limits between which this formula is appUetble.
NoU. It follows from the general theorem that the root y, eo oiw oreda e a
function of x and a, may be repreeented as a doublo Mvlee anrnofod a oc nwlit
to powers of x and a. Thia aeriea can be obtaiaod Iff wplMlaC ••ch ol iko
coefficients An by its development in power* of a. HoDOt the amim (M) moj
be differentiated term by term with reepeet to o.
Examples. 1) The equation
(63) y = a + ?(»«-.l)
406 SPECIAL SERIES [IX, §190
hM one root wliich is equal to a when x = 0. Lagrange's formula gives the
following development for that root :
(W)
xY d«-i (a2 - l)n
1 . 2 . . • n \2/ da^-i
On the other hand, the equation (63) may be solved directly, and its roots are
1 . 1
« = - ± - VT^2ax + a;2.
" X z
The root which is equal to a when jc = is that given by taking the sign --.
Differentiating both sides of (64) with respect to a, we obtain a formula which
differs from the formula (32) of § 184 only in notation.
2) Kepler's equation for the eccentric anomaly m,*
(66) u = a + e sin M ,
which occurs in Astronomy, has a root u which is equal to a f or e = 0. Lagrange's
formula gives the development of this root near e = in the form
e* d , . - , . , e" d^-Wsin^a)
(66) u = a-h esma + — - — (sm^a) + • • • + — j— H + ' * ••
1.2 da 1.2. -n da»-i
Laplace was the first to show, by a profound process of reasoning, that this
aeries converges whenever e is less than the limit 0.66274^8 • •
190. Inversion. Let us consider a series of the form
(67) y = 01X4-02x2 + .. • + a„a;» + ..-,
where ai is different from zero and where the interval of convergence is(— r, + r).
If y be taken as the independent variable and x be thought of as a function of y,
by the general theorem of § 187 the equation (57) has one and only one root which
approaches zero with y, and this root can be developed in a power series in y :
(68) X = 6iy + biV^ + ftgy^ + • • • + ft^y" + . • • .
The coefficients 6i, 62, 63 , • • • may be determined successively by replacing x in
(67) by this expansion and then equating the coeflficients of like powers of y.
The values thus found are
a, a\ aj
The value of the coefficient bn of the general term may be obtamed from
Lagraoge's formula. For, setting
^(x) = ai + a2X + . • . + OnX^-'^ + • . .,
tlM equation (67) may be written in the form
^(x)
*Bm p. 918, Ex. 19; and Ziwkt, Xlements of TTieoretical Mechanict, 2d ed.
p.aB6.— Teams.
IX, 5 191] HEAL ANALYTIC FL.NLllU^b 407
and the development of the root of thto eqaatioo whleh appraiebM tmo villi 9
is given by Lagrange*! formula in the form
%(0)
where the subscript indicatea that we are to lei s b after
indicated differentiations.
The problem just treated haa
191. Analytic functions. In the future we shall say that a foii»>
tion of any nuinber of variables x, y, s, ••• is analytic if it can be
developed, for values of the variables near the point x^, f^, *%t "*
in a power series arranged ai^cording to increasing powers of
X — Xq, y — i/of « — «o> • • • which converges for sufficiently small
values of the differences x — Xo, • • . The values which x,, y,, «,, • •
may take on may be restricted by certain conditions, but we shall
not go into the matter further here. The developments of the pree>
ent chapter make clear that such functions are, so to speak, inter-
related. Given one or more analytic functions, the operations of
integration and differentiation, the algebraic operations of multipli>
cation, division, substitution, etc., lead to new analytic functions.
Likewise, the solution of equations whose left-hand member is ana-
lytic leads to analytic functions. Since the very simplest functions,
such as polynomials, the exponential function, the trigonometrie
functions, etc., are analytic, it is easy to see why the first funetiocis
studied by mathematicians were analytic. These functiona are still
predominant in the theory of functions of a complex variable and ta
the study of differential equations. Nevertheless, despite the fuodap
mental importance of analytic functions, it must not be forgotten
that they actually constitute merely a very particular group among
the whole assemblage of continuous functions.*
192. Plane curves. Let us consider an arCi4B of a plane corva.
We shall say that the curve is analytic along the art A B it the
coordinates of any point M which lies in the neighborhood of any
fixed point iUy of that arc can be developed in power series arranged
according to powers of a parameter t — t^,
^ ^ ^y=^(0 = yo-f *i(<-^)+^(<--g« + -.+A.(f-f,)-+ .
which converge for sufficiently small values of f — <,.
• In the second volume an example of a n on a n a l yHc fteetlo« wHI heglvM, aUef
whose derivatives exist thronghoot an Interval (a, 6).
408 SPECIAL SERIES [IX, §192
A point 3/0 will be called an ordinary point if in the neighbor-
hood of that point one of the differences y — 2/0 ^ — ^q can be
represented as a convergent power series in powers of the other.
If, for example, y — Uo can be developed in a power series in
(60) y-yo = cxip^ - x,)-\-<^%{^ -x,y+-" + c^{x - x,y + ...,
for all values of x between Xf^ — h and x^ + h, the point {xq, y^) is
an ordinary point It is easy to replace the equation (60) by two
equations of the form (59), for we need only set
iX = Xq-{- t — tQ,
(^^> ?y = yo + Ci(^-A.)4---- + c„(^-g«+....
If Ci is different from zero, which is the case in general, the equa-
tion (60) may be solved for a; — a^o in a power series in y — y^ which
is valid whenever y — y^'is sufficiently small. In this case each of
the differences x — Xq, y — Vq can be represented as a convergent
power series in powers of the other. This ceases to be true if c^ is
zero, that is to say, if the tangent to the curve is parallel to the
x axis. In that case, as we shall see presently, cc — x^ may be devel-
oped in a series arranged according to fractional powers of y — y^'
It is evident also that at a point where the tangent is parallel to
the y axis x — x^ can be developed in power series in y — y^^ but
y — yo cannot be developed in power series ua x — x^.
If the coordinates (a:, y) of a point on the curve are given by the
equations (59) near a point Mq, that point is an ordinary point if
at least one of the coefficients ai, b^ is different from zero.* If a-i
is not zero, for example, the first equation can be solved for t — t^
in powers of x — a^o, and the second equation becomes an expansion
of y — yo ^ powers oi x -^Xq when this solution is substituted for
t - to.
The appearance of a curve at an ordinary point is either the cus-
tomary appearance or else that of a point of inflection. Any point
which is not an ordinary point is called a singular point. If all
the points of an arc of an analytic curve are ordinary points, the
arc is said to be regular.
• Thii condition Is iufficient, but not necessary. However, the equations of any
curre, near an ordinary iM)lnt Mq, may always he written in sucli a way that ai and
6i do not both vanlnli, by a auitable choice of the parameter. For this is actually
Moomplisbed in wiuatlous ((il). See also neeond footnote, p. 409. —Trans.
IX, 5l»3] Kt.vL AJSALllIt; FUNCTIONS
If each of the ooeffioients oi and 6| U lero, but a,, for exaiiipU,
is different from zero, the first of equatioot (59) may be writleo is
the form (x - Xo)* = (^ - Q^a^ 4- a, (<-<,)+. . .]*, whare the rifhi-
liand member is developable according to powera ot t — t^. HeuoB
t — Iq is developable in powers of (x — x,)*, and if < — /, in the
second equation of (59) be replaced by that deyelopment, we obtain
a development for y — y^ in powers of (x — a^)*:
y - yo = <h(aJ - «•)+«!(« - a!,)'+ «•(« - aw)«+ ••.
In this case the point (x^, y^ is usually a cusp of the fint kind.*
The argument just given is general. If the derelopmeot of
a; — Xo in powers of ^ — ^o begins with a term of degree ih y — |^
can be developed according to powers of (x — x^. The appearaooe
of a curve given by the equation (69) near a point (ae^, y,) is of
one of four types : a point with none of these peculiaritiea, a point
of inflection, a cusp of the first kind, or a cusp of the aeoood kind.*
193. Skew curves. A skew curve is said to be analytic aUng an art
AB if the coordinates x, y, z of b. variable point M can be dereloped
in power series arranged according to powers of a parameter < — («
j-x = xo -I- ax(^ - <o) + ••• -H «.(< - <•)■ + •••»
(62) \y = yo+f>,(t-to) + "+ b,(t - ^)' + ••.,
U = ;jJo + ci(^ - g -f- ... 4- c.(< -<,)•+ ...,
in the neighborhood of any fixed point M^ of the arc. A point
Mq is said to be au ordinary point if two of the three differences
X — Xq, y — yof ^ — ^0 ^^^^ ^ developed in power series arranged
according to powers of the third.
It can be shown, as in the preceding paragraph, that the point
Mq will surely be an ordinary point if not all three of the ooeiBeieota
ai,bi, Ci vanish. Hence the value of the parameter t for a lingular
point must satisfy the equations f
dt ^' dt ^* dt ^'
• For a cusp of the first kind the tangrat Um bttwwa Ik* two bcaaohM. Vbr a
cusp of the second kind both branches lie on tho mmm aldo of tko laagoM. TW
point is an ordinary point, of coane, U tho eoofloionti of tbo
happen to be all zeros. —Trawi.
t These conditions are not sniHeient to mako tho polat Jf«, whieh <
a value (q of tho parameter, a stngalar point whoa a polat if of tho oarro aoar Ms
corresponds to several values of t which approaeh !§ as if ayptoaohtt J^. 8eah Is
the case, for example, at the origin oo tho cnnro il s fln s d hj tho oqaatkas « ■ ^.
410 SPECIAL SERIES [IX, §l&4
Let x^f yof «o be the coordinates of a point Mq on a skew curve T
whose equations are given in the form
(63) F(x, y, «) = 0, Fi (x, y, z) = 0,
where the functions Fand F^ are power series m x — Xq, y — y^, z — z^.
The point Mq will surely be an ordinary point if not all three of
the functional determinants
Z)(F, F,) ^ D(F, F,) ^ D(F, FQ
I>(x, y) ' D{y, z) ' D(z, x)
vanish simultaneously at the point aj = ccq, y = yo, z = Zq. For if
the determinant D(F, Fi)/D(x, y), for example, does not vanish at
3/o, the equations (63) can be solved, by § 188, for x — Xq and y — y^
as power series in z — Zq.
194. Surfaces. A surface S will be said to be analytic throughout
a certain region if the coordinates x, y, z of any variable point M
can be expressed as double power series in terms of two variable
parameters t — t^ and u — Uq
«o = «io(^ — ^o) + aoi(^ — ""o) +
)) + •••,
») + •••,
in the neighborhood of any fixed point Mq of that region, where
the three series converge for sufficiently small values of t —t^ and
M — Mq. a point Mq of the surface will be said to be an ordinary
point if one of the three differences x — Xq, y — y^j z — Zq can be
expressed as a power series in terms of the other two. Every point
Mq for which not all three of the determinants
^(y> ^) ^ D(z, x) ^ D(x, y)
D{tj u)' D(t, u)' D(t, u)
vanish simultaneously is surely an ordinary point. If, for exam-
ple, the first of these determinants does not vanish, the last two of
the equations (64) can be solved for t — tQ and u — Uq, and the first
equation becomes an expansion of x — Xq in terms of y — yQ and
« — «o upon replacing t — to and ti — Uq by these values.
Let the surface S be given by means of an unsolved equation
^(*> y» *) = 0> and let Xq, y^, Zq be the coordinates of a point Mq
of the surface. If the function F(x, y, z) is a power series in
* ~ ^» y — yo» * — *o> and if not all three of the partial derivatives
BF/di^f dF/dy^, dF/dzo vanish simultaneously, the point M^ is surely
an ordinary point, by § 188.
{x — Xq = aiQ(t — to) 4- aoi(^ — ""o)
y - yo = *io(^ - ^o) + *oi {u - Wo)
z- Zo = c^o(t - fo) + ^'oi (w - '^^o)
lX,fl88j TRIGONOMETRIC SERIES 411
Note. The definition of an ordinary point on a cuire or 00 a ttir.
face is independent of the choice of axes. For, let M^ (2^, y„ l^) be aa
ordinary point on a surface S. Then the ooftrdinatee of any netgfa.
boring point can"^ be written in the form (64), where not all three of
the determinants D{y, z)/D(t, ti), Z)(«, x)/D{t, 11), Z)(x, y)/D{t, «)
vanish simultaneously for t^t^.usxti^. Let us now aelect any new
axes whatever and let
A' = aia; + Ay 4-yi« + ^,
K=a,x4-Ay + y,«4-«„
Z'!=ar,aj-|- Ay-|-y,« + S,
be the transformation which carries x, y, x into the new codrdinatee
A', }', Z, where the determinant A = /)(A', K, X)/D{x, y, «) is differ-
ent from zero, lieplacing a-, y, z by their developments in series
(64), we obtain three analogous developments for JT, K, iT ; and we
cannot have
D(t, u) D{ty u) D{t, u)
for t=itQ, u = Uq^ since the transformation can be written in the form
x = /l,A'-f Z?, r-f- C\Z -k-Dx,
y = yi,A' + /<,r+r,z + ^,
and the three functional determinants involving X, Y, Z eannoi
vanish simultaneously unless the three involving ar, y, « alao Taniah
simultaneously.
IV. TRIGONOMETRIC SERIES MISCELLANF.ol'S SERIKS
195. Calculation of the coefficients. The series which we shall study
in this section are entirely different from those studied above.
Trigonometric series appear to have been first studied by D. Ber>
noulli, in connection with the problem of the stretobed string. The
process for determining the coeftl«i»*iiLs. which we are about to irivf.
is due to Euler.
I^t f{x) be a function defined in the interval (a, A). Wc shall
first suppose that a and b have the values — w and + w, respec-
tively, which is always allowable, since the subctitutioo
.?•«»•—
412 SPECIAL SERIES [IX, §196
reduces any case to the preceding. Then if the equation
(6S)Jlz) = ^-{'(aiC08x + bi8mx)-\ [•(a^cosmx-\-b„8mmx)-\
Z
holds for all values of x between — ir and + tt, where the coefficients
<*o» «i> ^i» •••>««> *iii> •• • are unknown constants, the following device
enables us to determine those constants. We shall first write down
for reference the following formulae, which were established above,
for positive integral values of m and n :
I sin mx rfx = ;
I t087nxdx = {Sy if Wt>^0;
I cos -mx cos wa; dx
X"^ ' cos (w — w) X -f cos (m 4- n)x , ^ . -
^^ ^ — -z ^' ^— cfx = 0, 11 m ^ n;
/"*"', , C^"" l-\-G0s2mx , .n ^
I co8*wia;tfa;= I ^ dx =7r, if m=?^0;
I sin mx sin no; <fa;
r cos (m — n)a; — cos Cm 4- n)x , ^ . „
= / ^^ ^ — 2 ^ — ^^— ^<fo; = 0, if m^Ti;
X"*"' . a , r"*"' 1 — cos2ma5 ,
sm^'Twajdaj = j ^ (to =7r, if w =?t= Oj
I sin mx cos Ttx cKx
X sin (m + yi)a; + sin (m — n)x j
2 dx = 0.
Integrating both sides of (65) between the limits - tt and + tt,
the right-hand side being integrated term by term, we find
(66)
£'/(.)*.=|£'<^
which gives the value of a^. Performing the same operations upon
the equation (66) after having multiplied both sides either by cos mx
IX.} 190] TRIGONOMETRIC SERIES 418
or by sin mxj the only tenn on the right whoie integral
and + TT is different from aero is the one in ooe'm« or in ein'eia.
Hence we find the formulas
/ /(x) cosmxdx =i Tra., / f(x)
mxdxwB vi^y
respectively. The yalues of the ooefflcients may be etiiiinljlei]
follows :
(67)
r 1 r^' I /* •
1 r*'
b^ = — I /{a) sin ma da.
The preceding calculation is merely formal, and therefore tent^
tive. For we have assumed that the function /(i) can be developed
in the form (65), and that that development oonrergee uniformly
between the limits — w and + tt. Sinoe there is nothing to prove,
a priori, that these assumptions are justifiable, it is essential that
we investigate whether the series thus obtained converges or noL
Replacing the coefHcients a^ and bt by their values from (67) and
simplifying, the sum of the first (m + 1) terms is seen to be
h f /wri+co8(^-*)+«^2(*-*)+-+***"»(*~')i^-
But by a well-known trigonometric formula we have
. 2»+l
- -h cos a -h cos 2a H h cos smi*« " — »
2sin|
whence
• 2m-|-l, .
1 r*' "° — 2~^''"*)
^^- 2sin^^
or, setting a = a; -f 2y,
(68) 5... = -J _/[« + 2y) \i„y '' <'»■
t
The whole question is reduced to that of finding the limit of Ihit
siun as the integer m increases indefinitely. In order to stodj this
question, we shall assume that the function /[x) salis6et «!• fol-
lowing conditions :
414 SPECIAL SERIES [IX, §196
1) The function f(x) shall be in general continuous between — ir
and -f TT, except for a finite number of values of x, for which its value
may change suddenly in the following manner. Let c be a number
between — tt and -|- tt. For any value of c a number h can be found
such that fix) is continuous between c — h and c and also between
c and c-\- h. As c approaches zero, f{c + c) approaches a limit which
we shall call /(c -f 0). Likewise, f(c — e) approaches a limit which
we shall call f(c — 0) as c approaches zero. If the fimction f(x)
is continuous for a; = c, we shall have /(c) =/(c + 0)=/(c — 0). If
f(e -f 0) ^ f{c — 0)jf(x) is discontinuous for x = c, and we shall agree
to take the arithmetic mean of these values [/(c + 0) +/(c — 0)]/2
for /(c). It is evident that this definition of /(c) holds also at points
where f(x) is continuous. We shall further suppose that /(— tt + c)
and /{it — c) approach limits, which we shall call /(— tt + 0) and
/(tt — 0), respectively, as c approaches zero through positive values.
The curve whose equation is y =f{^) must be similar to that of
Fig. 11 on page 160, if there are any discontinuities. We have
already seen that the function f(x) is integrable in the interval from
— TT to 4- TT, and it is evident that the same is true for the product
oif(x) by any function which is continuous in the same interval.
2) It shall be possible to divide the interval (— tt, + tt) into a
finite number of subintervals in such a way that/(£c) is a monoton-
ically increasing or a monotonically decreasing function in each of
the subintervals.
For brevity we shall say that the function f{x) satisfies Dirichlefs
conditions in the interval (— tt, + tt). It is clear that a function
which is continuous in the interval (— tt, + tt) and which has a
finite number of maxima and minima in that interval, satisfies
Dirichlet^s conditions.
196. The integral J^ f (x) [sin nx/sin x] dx. The expression obtained
for 5^^, leads us to seek the limit of the definite integral
X'
sin nx
. /(«) — dx
fo sm a;
88 n becomes infinite. The first rigorous discussion of this ques-
tion was given by Lejeune-Dirichlet.* The method which we shall
employ is essentially the same as that given by Bonnet. f
• CfrttU's Journal, Vol. IV, 1829.
t MHnoiru det tavanta strangers publics par I'Acaddmie de Belgique, Vol. XXIIL
DC, $198] TRIGONOMETRIC SERHES 416
Let us first consider the integral
(69) y=jrV)?i^^.
where A is a positive number less than w, and 4(x) U a
which satisfies Dirichlet's conditions in the interval (0, A). If 4(«)
is a constant C, it is easy to find the limit of J, For, iiitlim y ■* <»b
we may write
'-X T*"'
and the limit of / as n becomes infinite is Cw/2, by (99), f 176.
Next suppose that ^(2^) is a positive monotonically deerMtfaif
function in the interval (0, h). The integrand ohanget tigii for
all values of x of the form kw/n. Henoe J may be written
where
tti. =
(* + !)»
sinn^
X " "^^'^
dx
and where the upper limit A is suppoeed to lie between wtw/m and
(7n -f- l)7r/7i. Each of the integrals u^ is less than the preoeding.
For, if we set na; = A*7r 4- y in u^, we find
JC'^ / y-h Anr \ siny .
and it ia evident, by the hypotheses regarding ^('), that this inl»>
gral decreases as the subscript k increases. Henoe we shall haT«
the ecjuations
/ = tto - (t«i - V,) - (m, - k«) ,
y = t£„ - w, -I- (14 - u.) 4- (|^ - «») + • • • .
which show that J lies between «, and ^ — Ut, It follows thai J it
a positive number less than u,, that is to 9Mj, lest than the intflfiml
X
^ , . sm IMP .
-^
But this integral is itself less than the integral
where .1 denotes the value of the definite integral ^*[(«n fi/y]^
416 SPECIAL SERIES [IX,§19«
The same argument shows that the definite integral
r* . , ^ sin na; ,
where c is any positive number less than h, approaches zero as n
becomes infinite. If c lies between (t — l)'7r/n and iw/n, it can be
shown as above that the absolute value of J^ is less than
smnx
<f>{x) dx
and hence, a fortiori, less than
<^(c) (ijr _\. \n} IT ^ 27r <^(c)
c Vw / iir n n c
Hence the integral approaches zero as n becomes infinite.*
This method gives us no information if c = 0. In order to dis-
cover the limit of the integral /, let c be a number between
and hj such that <^(a;) is continuous from to c, and let us set
^{x) = ^(c) 4- ^{x). Then xj/ix) is positive and decreases in the
interval (0, c) from the value <^(+ 0) — <^(c) when a; = to the
value zero when a; = c. If we write / in the form
=*(c)jr'5i
sm nx
dx -h
X\ . . sin Tw; - ,
^(aj) — :;; — dx -\-
and then subtract (7r/2)<^(+ 0), we find
X *(^) -
sinTiaj
dx
(70)
Ja ^ Jc
sinn«c
In order to prove that J approaches the limit (ir/2) <^(-f- 0), it will
be sufficient to show that a number m exists such that the absolute
• ThU result m»y be obtained even more simply by the use of the second theorem
of the mean for integrals (§75). Since the function ^(x) is a decreasing function,
that formula gives
ABd Um right-hand member evidently approaches zero.
IX, $196] TRIGONOMETRIC SERIES 417
value of each of the terms on the right U leM than a pMMtifBfld
positive number (/4 when n is greater than m. By the remark
made above, the absolute value of the integral
/■
is less than A ^(-}- 0) = i4 [^(+ 0) — ^(f)]. Since ^{x) approachm
<^(+ 0) as X approaches zero, e may be taken so near to zero thai
A [<^(+ 0) - <^(c)] and (•»r/2)[^(+ 0) - ^c)] are both less than t/4.
The number c having been chosen in this way, the other two terma
on the right-hand side of equation (70) both approaeh laro as a
becomes infinite. Hence n may be chosen so large that the abeo-
lute value of either of them is less than «/4. It followt that
(71) limy = f^+0).
We shall now proceed to remove the various restrictions which
have been placed upon <f>{x) in the preceding argument. If 4(') >*
a monotonieally decreasing function, but is not always poattive, the
function ^(a;) = <^(a;) + C is a positive monotonieally deereaatng fime-
tion from to A if the constant C be suitably chosen. Then the
formula (71) applies to ^(^). Moreover we may write
Jr\, ^ sinnar , f*. aintur, ^ f * sin nx ^
and the right-hand side approaches the limit (w/2)^+0) — (w/2)Cp
that is, (7r/2)<^(+0).
If <f>(x) is a monotonieally increasing function from to A, — 4(ar)
is a monotonieally decreasing function, and we shall have
XV)^^=-X'-*^'^
-Jx.
Hence the integral approat^hes (^/2)o
Finally, suppose that ff>{x) is any fu: - > — Dirich-
let's conditions in the interval (0, A). Then the interval (0, k)
may be divided into a finite numbt^r of subintervaU (0, «), (a, k),
(P> c\ • • •, (^» A)' in ^^^ of ^'^^'^^* ^(') ** * monotonieally increasing
or decreasing function. The integral from to a approaches the limit
(7r/2)<^(4- 0). Each of the other integrals, which are of the type
H^f\(
^ Sin nx ,
x)——dx,
418 SPECIAL SERIES [IX, §197
approaches zero. For if <l>(x) is a monotonically increasing function,
for instance, from a to 6, an auxiliary function ^(ic) can be formed
in an infinite variety of ways, which increases monotonically from
to ft, is continuous from to a, and coincides with <f}(x) from a to b.
Then each of the integrals
X" , ^ sin no; , r^ sinnx
approaches ^(-f- 0) as w becomes infinite. Hence their difference,
which is precisely H, approaches zero. It follows that the formula
(71) holds for any function <t>{x) which satisfies Dirichlet's condi-
tions in the interval (0, h).
Let us now consider the integral
(72) ^=X/(-)l^'^' 0<A<,r,
where f(x) is a positive monotonically increasing function from
to A. This integral may be written
-X'[/<-)ii^]^*-
and the function <l>(x) = /(x) ic/sin a; is a positive monotonically
increasing function from to h. Since /(+ 0) = <^(+ 0), it follows
that
(73) lim/ = |/(+0).
This formula therefore holds if f(x) is a positive monotonically
increasing function from to h. It can be shown by successive
steps, as above, that the restrictions upon f(x) can all be removed,
and that the formula holds for any function f(x) which satisfies
Dirichlet's conditions in the interval (0, h).
197. Fourier series. A trigonometric series whose coefficients are
given by the formulae (67) is usually called a FouHer series. Indeed
it was Fourier who first stated the theorem that any function arbi-
trarily defined in an interval of length 27r may be represented by a
series of that type. By an arbitrary function Fourier understood
a function which could be represented graphically by several cur-
vilinear arcs of curves which are usually regarded as distinct curves.
We shall render this rather vague notion precise by restricting our
discussion to functions which satisfy Dirichlet's conditions.
IX.§1!)7] TRIGONOMETRIC 8BRIE8 419
In order to show that a function of this kind can be Mipw Hwu liMi
by a Fourier series in the interval {— w, -^ w), we mttst find t^
limit of the integral (68) as m becomes infinite. Lei us divide
this integral into two integrals whose limite of integrataon mm
and (IT - a-)/2, and - (ir + x)/'J and 0, respeetively, and let nt
make the substitution y =a — z in the seoood of theee tntagrml^
Then the formula (68) becomes
frJo '^^ ' sm«
When X lies between — ir and + w, (w — x)/2 and (w -♦- ar)/2 both
lie l)etween and tt. Hence by the last article the right-hand
side of the preceding formula approaches
1 [E^(. + 0) + lA^ - 0)] = 2L' >o)±2][£^
as m becomes infinite. It follows that the series (65) oooTerges and
that its sum is /(a;) for every value of x between — w and + ».
Let us now suppose that x is equal to one of the limits of the
interval, — ir for example. Then ^'.^.j may be written in the fom
The first integral on the right approaches the limit/t—ir + •)/!.
Setting y = TT — « in the second integral, it takes the form
^Jo •
Z /(^
,. gin(2m.H)s^^
sins
whicli approaches /(7r - O)/!'. Hence the sum oi ihe ir..
series is [./(tt - 0) -\- f{- w + 0)]/2 when ar = - w. It . • •
that the sum of the series is the same when » «+ w.
If, instead of laying off x as a length along n strmight iinr, ww
lay it off as the length of an arc of a unit circle, connling in Um
420 SPECIAL SERIES [IX, §197
positive direction from the point of intersection of the circle with
the positive direction of some initial diameter, the sum of the series
at any point whatever will be the arithmetic mean of the two limits
approached by the sum of the series as each of the variable points
m' and m", taken on the circumference on opposite sides of m,
approaches m. If the two limits /(— tt + 0) and /(tt — 0) are
different, the point of the circumference on the negative direction
of the initial line will be a point of discontinuity.
In conclusion, every function which is defined in the interval
(— TT, 4- tt) and which satisfies Dirichlefs conditions in that inter-
vcU may be represented by a Fourier series in the same interval.
More generally, let f(x) be a function which is defined in an
interval (a, a -f 27r) of length 27r, and which satisfies Dirichlet's
conditions in that interval. It is evident that there exists one and
only one function F(x) which has the period 27r and coincides with
f(x) in the interval (a, a + 27r). This function is represented, for
all values of x, by the sum of a trigonometric series whose coeffi-
cients a^ and b^ are given by the formulae (67):
1 r^" 1 r^""
dm — - j F{^) cos mxdx, b„= - j F(x) sin mx dx.
The coefficient a^, for example, may be written in the form
am = ~ I ^M COS mxdx-{-- I F(x) cos mx dx ,
where a is supposed to lie between 2A7r — tt and 2A7r + tt. Since
F(x) has the period 27r and coincides with f(x) in the interval
(a, a 4- 27r), this value may be rewritten in the form
(74)
<*« = -/ f{x) COS m^dx-\- \ f{x) cos mx dx
= — I f{x) COS mx dx.
^ c/ar
Similarly, we should find
-« /»<r-flir
(75) h^=i- i f{x)%mmxdx.
Whenever a function /(x) is defined in any interval of length 27r,
the preceding formulae enable us to calculate the coefficients of its
development in a Fourier series without reducing the given interval
to the interval (- tt, -f tt).
IZ.flW] TRIGONOMETRIC SERIES 421
191. ExampUt. 1) Let tu And a Foaiter ttrlti wboat mm to - 1 i
« < z < 0, and +1 for < x < + «. TU ionBBtai (67) glf* ih* vsIims
1 /•• 1 /»»
00 = - / -<te + - I dacsO,
\. = i r*-«inmxdx + i /*'■«.. -..^ - > - «»"*^ - «»(- »^
If m is eyen, b^ is zero. If m la odd, 6. is 4/m«. MulUplylnf all tba
cients by ^/4, we see that the sum of the
(76) Binx sinSx rin(«m-H)«
+ 1
is - 9r/4 for - )r < z < 0, and + fc/A for < x < r. Tba pofaH « k Ohm polst
of discontinuity, and the sum of the leriea is zero when x = 0, as it rtpmM ba.
More generally the sum of the series (76) is «/4 when slnx ia poilU?*, r'i
when sin x is negative, and zero when sin z = 0.
The curve represented by the equation (76) ia oompoaed of an tnflBlla namber
of segments of length n of the straight lima y = ± r/4 and an Infialla ^'^m-
ber of isolated points (y = 0, z = kit) on the z axis.
2) The coefficients of the Fourier derelopment of z in the intertai from 10
2n are
ao= - / zdz = 2*,
1C Jo
1 f" , rz sin mz-l«» . 1 /•••. . ^
a^=-l xcosmzdz= |___ J^ + _ j^ *im.*.0.
. 1 r*' 1 ^ rzcoamx-l«» . 1 /•»» . S
6,n=-/ zsinmzdz = - — — — — + — - / ooaiwAa
Hence the formula
m\ z _ y ainx sinax atnSx
^^ 2"2 1 2 ~ S
is valid for all values of z between and Sir. If wa aM y aqoal lo tka avfM tm
the right, the resulting equation repreanta a eorraooapoiaddf ■• tetidlaBaa-
ber of segments of straight lines parallel to y = s/1 and aft laSnlto MUBbw of
isolated points.
NoU. If the function f{x) deflnad in tha intarral (- v, -t- a) Is mm, IkM la
to say, if /(- z) =/(z), each of tte aoaOolaBta tb to aaro, aiaoa it to
ainaixdii.
J*/l(z)ainmxdx = -^/[x)
Similarly, if /(z) is an odd fanettOB, that to. If /[ - x) = -/tx). earh of the
coefficients a« i^ zero. Including d*. A fooetioB /fx) whieh to daOned oolj tn
422 SPECIAL SERIES [IX, §199
the interval from to »r may be defined in the interval from - tt to by either
of the equations , . ^, .
f{-x)=f{x) or /(-x) = -/(x)
If we choose to do so. Hence the given function /(x) may be represented either
by a series of cosines or by a series of sines, in the interval from to ;r.
199. Expansion of a continuous function. Weierstrass' theorem. Let /(x) be a
function which is defined and continuous in the interval (a, 6). The following
remarkable theorem was discovered by Weierstrass : Given any preassigned posi-
tive number «, a polynomial P(x) can always be found such that the difference
/(x) _ p(x) is less than c in absolute value for all values of x in the interval (a, b).
Among the many proofs of this theorem, that due to Lebesgue is one of the
simplest* Let us first consider a special function \p{x) which is continuous in
the interval (- 1, + 1) and which is defined as follows : \p{x) = for - 1 <x < 0,
f (x) = 2ifcx for < X < 1, where A; is a given constant. Then yf/{x) = (x + | x |) A;.
Moreover for - 1 < x < + 1 we shall have
|xi = vr^(i-x2),
and for the same values of x the radical can be developed in a uniformly con-
vergent series arranged according to powers of (1 - x^). It follows that |x|, and
hence also ^(x), may be represented to any desired degree of approximation in
tne interval (- 1, + 1) by a polynomial.
Let us now consider any function whatever, /(x), which is continuous in
the interval (a, 6), and let us divide that interval into a suite of subintervals
(oo, ai), (ai , az), • • • , (a„-i , a„), where a = ao < ai < az < • • • < On-i < a„ = 6,
in such a way that the oscillation of f{x) in any one of the subintervals is less
than c/2. Let L be the broken line formed by connecting the points of the
curve y = /(x) whose abscissae are ao , ai , ag , • • • , &. The ordinate of any point
on L is evidently a continuous function 0(x), and the difference f{x) — <f>{x) is
less than e/2 in absolute value. For in the interval (a^a-i, a^), for example,
we shall have
fix) - 0(x) = [fix) -/(a^-i)] (1 - ^) + [fix) -fia^)]0,
where x — 0/4_i = tf(a,i — o^_i). Since the factor d is positive and less than
unity, the absolute value of the difference f — <f> is less than c(l — ^ + 0)/2 = c/2.
The function 0(x) can be split up into a sum of n functions of the same type as
f (x). For, let ^0, -4i, ^2, • • • , ^n be the successive vertices of L. Then 0(x)
is equal to the continuous function ^i (x) which is represented throughout the
interval (o, 6) by the straight line AqAi extended, plus a function 0i(x) which
is represented by a broken line AoAi- •- Ai, whose first side AqAi lies on the
X axis and whose other sides are readily constructed from the sides of L. Again,
the function <pi (x) is equal to the sum of two functions ^2 and 02 » where ^2 is
zero between Oo and ai, and is represented by the straight line A1A2 extended
between ai and 6, while 02 is represented by a broken line AoAi'Ai- • • An whose
first three vertices lie on the x axis. Finally, we shall obtain the equation
^ = ^i + ^1 + • • • + ^H , where ^< is a continuous function which vanishes
between Oq and ot-i and which is represented by a segment of a straight line
* Bulletin dea sciences matMrnatiquen, p. 278, 1898.
lX,§aOO] TRIGONOMETRIC SKRIKS 42|
between o^.i and 6. it we tuen maxe uie suimiuiuuuii X b «k 4. i^ wImto m
and 71 aru suitably choten numben, the fonsikMi f #(») Btj bt ditetd Is Ite
interval (- 1, -f 1) by tbe equation
f,(x) = t<Jr + |X|).
and hence it can be repreaented by a polynomial with any rtMJml dtfiM oC
approximation. Since each of the functiona f<(z) can be repiMMiad la Ik*
interval (a, 6) by a polynomial with an error le« than i/Sa, It la
sum of these polyuomiala will differ from/(x) by leea than «.
It foUowH from the preceding theorem that any /untUom /{x) wkkk <■
nous in an interval (a, 6) inay be rtprtteniti by an ii^niU mrim ^ jrtrfrnirmiah
Mohich converges uniformly in that interval. For, let <i , c«, .,<.,.•. bt a 1
of positive numbers, each of which is less than the preceding, where ^1
zero aa n becomes infinite. By the preceding theorem, corraqModlBf to caieii of
the c*8 a polynomial P, (x) can be foond anch that the dUlereoee /(«) ~ Pi{g)lt
leas than e, in absolute value throughout the interval (a, 6). Then tbe amim
Pi(x) + [P,(x) - P,(x)] + .. . + [P.(x) - P.-i(*)) + •••
converges, and its Hum is/(x) for any value of x iiiaide the interval (a, 6). For
the sum uf the tirst n terms is equal to P.fx), and the differeooe /(x) — d^, whleh
is less than Cn , approaches zero as n becomes infinite. Moreover th* ■artea cem-
verges uniformly, since the absolute value of the difference /(x) — 8m win be Icai
than any preaBsi<,'ned positive number for all valuea of n which eieead m eeftnln
fixed integer N, when x has any value whatever between a and 6.
200. A continuous function without a derivative. We ahall ooodudo thii ebaplor
by giving an example due to Weierstraas of a oontlnoona fnnctioo whlell does
not possess a derivative for any value of the variable whatever. Let 6 bt a poil-
tive constant less than unity and let a be an odd Inttfer. Then the ftnwrtf
F{x) defined by the convergent infinite aeries
(78) F(x) ='^b^coa{c^ xz)
is continuous for all values of x, ainct the teriea eoowftt unUotmif hi aaj
interval whatever. If the product ab la lea than unity, tht
hold for the series obuined by term-by-term dlflettB th l tfc wi,
tion F{z) possesses a derivative which ia ittelf a eontinnoat fkuelkm. Wt Anil
now show that the sUte of affairs is eteentially difftrtnt II tht
a certain limit.
In the first place, setting
M-l
S« = i V fr-:oos[a-ir(x + A)) - coe(«««x)},
■»■•»
ii, = 1 2^ 6- {cot [a- r(« + A)l - oot (•• w)) .
we may write
424 SPECIAL SERIES [IX, §200
On the other hand, it is easy to show, by applying the law of the mean to the
function coe(a"«'x), that the difference co8[a'»«'(x + h)] — coa {a"* tcx) is less than
iro" \h\in absolute value. Hence the absolute value of Sm is less than
m-l
X a"» o»» — 1
a**©" = Tf — - — — - >
ao — 1
and consequently also less than «r(a6)'»/(a6 — 1), if oft >1. Let us try to find a
lower limit of the absolute value of R„ when h is assigned a particular value.
We shall always have
a'»x = am + lm,
where a>, is an integer and fm Hes between — 1/2 and + 1/2. If we set
1. ^m Cm
a™
where c is equal to ± 1, it is evident that the sign of h is the same aa that of
C, and that the absolute value of h is less than 3/2a"'. Having chosen h in this
way, we shall have
a" ir{x ■\- h)z= a^-f^a"^ 7t{x + h) = a^-"^7t{am + fim) •
Since a is odd and e™ = ±1, the product a»-*»{am + em) is even or odd with
a^ -f- 1, and hence
cos[a»;r(x + A)] = (- l)«m+i.
Moreover we shall have
C08(a»;rx) = cos(a«-»«a'»;rx) = cos[a»-"»;r(aTO + ^m)]
= cos (a"-*" OTm ?r) cos (a»-»* ^^ tt) ,
or, since a"-"»a« is even or odd with a^
cos{a^jtx) = (-l)«i»cos(a'»-"»fm^).
It follows that we may write
Rn.=
(_!)«« + !
+ 00
2)&"[l+cos(a»-'»^,„;r)].
Since every term of the series is positive, its sum is greater than the first term, and
oonaeqaently it is greater than 6« since U lies between - 1/2 and + 1/2. Hence
or, since |A|<3/2a"',
If a and h satisfy the inequality
(80) a6>l + ^,
we shall have
2 / rv„ ^ 7r(a6)»
whence, by (79),
IX. Exs] EXERCI8E8 4S5
As m becomes infinite the expre«ion on tiM tttreae rifbt :
while the absolute valtte of h ftpproaehee mto. rninigpillj, -w
small < be choeen, an increment h can be found which to lea ihaa < la
lute value, and for which the absolute value of [F{z + A) - F{M)]/k aaaate aay
preaesigned number whatever. It follows thai if a and 6 aitiiCy the nlitlna flO),
the function F{x) possesses no derivative for any value of «
ixnciSM
1. Apply Lagrange^s formula to derive a develofmiat tai powws of s df Iteft
root of the equation y< = ay + z which to equal io a wbao • a 0.
2. Solve the similar problem for the equation f ^ a-^ t^*^ 8 0. Apply tkt
result to the quadratic equation a - te + cz* = 0. D«vtlop la powwi of e tkaft
root of the quadratic which approaches a/6 as c approoehss asro.
3. Derive the formula
4. Show that the formula
X
VI
holds whenever z is greater than - 1/2
6. Show that the equation
1 2z
x =
2.4V1 + W ^2.4.«Vl + W
2 l + ««
holds for values of z less than 1 in absolute value. What to the sum of the
when 1 z I > 1 ?
1 r nx ^ njn-l) / x V it(«-l)(ii>t) / « V 1
a-L a + x"^ 1 2 \o + xy"" l.a.S \«+«/ J
6. Derive the formula
(a + z)-
7. Show that the branches of the funetloB iiBflisaBd«
to and 1, respectively, when sinx = are do fl opa b to i&
powers of sin X :
1.2 1 . » . o . S
[Make use of the differential equation
which is satisfied by u = oosmx and by « s iin ms, whors y v ilncj
8. From the preceding fonnotos dodoos
cos (n arc COS x), da(«af«oai«).
CHAPTER X
PLANE CURVES
The curves and surfaces treated in Analytic Geometry, properly
speaking, are analytic cui'ves and surfaces. However, the geomet-
rical concepts which we are about to consider involve only the exist-
ence of a certain number of successive derivatives. Thus the curve
whose equation is y =f(x) possesses a tangent if the function f(x)
has a derivative f'(x) ; it has a radius of curvature if f'(x) has a
derivative f"(x) ; and so forth.
I. ENVELOPES
201. Determination of envelopes. Given a plane curve C whose
equation
(1) • f(x, y,a) =
involves an arbitrary parameter a, the form and the position of the
curve will vary with a. If each of the positions of the curve C is
tangent to a fixed curve E, the curve E is called the envelope of the
curves C, and the curves C are said to be enveloped by E. The
problem before us is to establish the existence (or non-existence) of
an envelope for a given family of curves C, and to determine that
envelope when it does exist.
Assuming that an envelope E exists, let (x, y) be the point of tan-
gency of E with that one of the curves C which corresponds to a cer-
tain value a of the parameter. The quantities x and y are unknown
functions of the parameter a which satisfy the equation (1). In
order to determine these functions, let us express the fact that the
tangents to the two curves E and C coincide for all values of a.
Let hx and By be two quantities proportional to the direction cosines
of the tangent to the curve C, and let dx/da and dy/da be the
derivatives of the unknown functions x = </>(«), y = \j/{d). Then a
neoessary condition for tangency is
dx dy
426
X,jaoiJ ENVELOl'Ks 4IJ
On the other hand, since a in equAtion (1) haa % eoutont Taloa for
the particular curre C considered, we shall have
which determines the tangent to C. Again, the two unknown fua^
tions X = ^(a), y = ^(a) satisfy the equatioo
yi*»y, «) = 0,
also, where a is now the independent variable. Uenot
or, combiniDg the equations (2), (3), and (4),
(5) lf-0.
The unknown functions x — ^(a), y = ^a) are aolatioof of thia eqa^
tion and the equation (1). Hence the equation of the enp^pe^ im
ease an envelope exists^ is to be found by eliminatiny the partm^Ur^
between the equations / = 0, df/ca = 0.
Let R{x, ^) = be the equation obtained hf eliminating a b e t wee a
(1) and (5), and let us try to determine whether or not this equatioQ
represents an envelope of the given curves. Let C, be the pariien*
lar curve which corresponds to a value a^ of the parameter, and lei
(^0) Vo) ^ t^6 coordinates of the point A/, of intemettOD of Um
two curves
(6) /(x,y,«o) = 0, ^ = 0.
The equations (1) and (5) have, in general, Bolutiona of the form
X = <^(a), y = ^(a), which reduce to z^ and y«, retpMltTelj, for
a = Oy. Hence for a = a^ we shall have
dxAda)^^ dyAda).
This equation taken in connection with the equation (3) ahovt
that the tangent to the curve r, coincides with the tangeoi to tlie
curve described by the point (x, y), at least unless df/dx and df/^
are both zero, that is, unless the point -V, is a singular point for the
curve Co. It follows that the equation /?(x, y) - reprmrmiM either
the envelope of the eurvei C or eUe the hens ^ iinpUmr points om
these curves.
428 PLANE CURVES [X,§202
1 his result may be supplemented. If each of the curves C has
one or more singular points, the locus of such points is surely a part
of the curve /?(a;, y) — 0. Suppose, for example, that the point (x, y)
is such a singular point. Then x and y are functions of a which
satisfy the three equations
f{x.y,a) = 0, ^ = 0, g = 0,
and hence also the equation df/da = 0. It follows that x and y
satisfy the equation R{Xj y) = obtained by eliminating a between
the two equations / = and df/da = 0. In the general case the
curve R(Xf y) = is composed of two analytically distinct parts,
one of which is the true envelope, while the other is the locus of
the singular points.
Example. Let us consider the family of curves
f{x,y,a) = y'-y' + {x-ay = 0.
The elimination of a between this equation and the derived equation
|f = -2(x-«) =
gives y* — y* = 0, which represents the three straight lines y = 0,
y = + l, y= — 1. The given family of curves may be generated
by a translation of the curve y* — y^ + a;^ = along the x axis.
This curve has a double point at the origin, and it is tangent to
each of the straight lines y = ± 1 at the points where it cuts the
y axis. Hence the straight line y = is the locus of double points,
whereas the two straight lines y = ± 1 constitute the real envelope.
202. If the curves C have an envelope E, any point of the envelope
is tht limiting position of the point of intersection of two curves of
the family for which the valvss of the parameter differ by an infini-
tesimal. For, let
(7) f(x, y, a) = 0, f(x, y, a -{- h) =
be the equations of two neighboring curves of the family. The
equations (7), which determine the points of intersection of the two
coryes, may evidently be replaced by the equivalent system
X, § 'M-2] ENVELOPES 4f9
the seoend of which reduces to d//da » m A approtehei »ro, ihal
is, as the second of the two curves approaches the first, Thi« pto^
erty is fairly evident geometrically. In Fig. 37, a, for injunca, thm
point of intersection ^V of the two neighboring curves C and C*
approaches the point of tangencj M na C ipproaohef Um enrrs C
Fia. 37, a FW. 37, 6
as its limiting positica. Likewise, in Fig. 37, &, where the pi von
curves (1) are supposed to have double points, the point of intersoc-
tion of two neighboring curves C and C ^>proaehes the point wber*
C cuts the envelope as C approaches C.
The remark just made explains why the locus of singular potots
is found along with the envelope. For, suppose that J\x^ y, a) is a
polynomial of degree m in a. For any point A/«(aE;t, y«) eboaaa at
random in the plane the equation
(8) • Ax^,y.,a)^0
will have, in general, m distinct roots. Through such a point theiv
pass, in general, m different curves of the given family. But if tha
point Mq lies on the curve /?(x, y) = 0, the equations
are satisfied simultaneously, and the equation (8) has a double root
The equation n(x, y) = may therefore be said to rsptaaspl tbo
locus of those points in the plane for which two of the eurrsf of
the given family which pass through it have merged into a single
one. The figures 37, a, and 37, A, show clearly the manner in wkieh
two of the curves through a given point merge into a single one as
that point approaches a point of the eunre if(«, ]f ) » 0| whotlior OS
the true envelope or on a lot'us t»f double points.
430 PLANE CURVES [X,§203
Note. It often becomes necessary to find the envelope of a family
of curves
(9) F{x, y, a, ^) =
whose equation involves two variable parameters a and 5, which
themselves satisfy a relation of the form <^(a, b) = 0. This case
does not differ essentially from the preceding general case, however,
for 6 may be thought of as a function of a defined by the equation
^ = 0. By the rule obtained above, we should join with the given
equation the equation obtained by equating to zero the derivative
of its left-hand member with respect to a :
da db da
But from the relation <^(a, 6) = we have also
d^ d<f> db _
da db da '
whence, eliminating db/da, we obtain the equation ^ ^ ^
^^"^ da db db da " "'
which, together with the equations F = and </> = 0, determine the
required envelope. The parameters a and b may be eliminated
between these three equations if desired.
203. Envelope of a straight line. As an example let us consider the equation
of a straight line D in normal form
(11) xcosa + ysinar -/(a) = 0,
where the variable parameter is the angle a. Differentiating the left-hand side
with respect to this parameter, we find as the second equation
(12) — xsina + ycosa — /(a) =0.
These two equations (11) and (12) determine the point of intersection of any
one of the family (11) with the envelope E in the form
. jgv ( X = f(a) cos a - f'{a) sin a ,
j y = f{a) sin a + f{a) cos a .
It is easy to show that the tangent to the envelope E which is described by this
point (z, y) is precisely the line D. For from the equations (13) we find
/14V ( dx = - [/{a) ■\-ria)] sinorda,
^ ' ldy= [/(a) 4- /"(a)] cos a dor,
whence dy/dx = — cot a , which is precisely the slope of the line D.
X.$203]
ENVKLOPE8
ttl
Moreover, If g denote the length of the an of ih» MvvloDe horn «•• a—j
point upon it, we hare, from (U), ^^ ^^ ^ "••
• = ± [//(a) da +/'(«) J.
Hence the envelope will be a curve which to eadly racUflable If we
for/(a) the derivative of a known function.*
As an example let ua set /(a) = / sin a coe a. Tkklng ysOaods.OMA.
cessively in the equation (11), we find (Fig. 88) O^ s /atoa OBmlcml
respectively ; hence AB = I. The required '
curve is therefore the envelope of a atraight
line of constant length /, whoee extremiUee
always lie on the two axes. The formuUe
(13) give in this case
« = isin»a, y = /coe«a,
and the equation of the envelope la
(;)-(0^=
which represents a hypocycloid with four
cusps, of the form indicated in the ti/^ure.
As a varies from to «'/2, the point of con-
tact describes the arc DC. Hence the length of the arc, counts fmm D, to
Flo. as
=X'«
alnaooaada
8/
— (
S
Let / be the fourth vertex of the rectangle determined bj OA tad OA, tDd Jf
the foot of the perpendicular let fall from / upon AB. Tliia, troa ll» tri-
angles AMI and APM, we find, snoceMlvely,
^3f = ^/coea = <oo8*a, AP = AMtina es leo^atHma.
Hence OP = OA - AP = t sin'a, and the point if to th« poiat ol
the line AB with the envelope. Moreover
hence the length of the are DM s ZBM/%.
• Each of the quantities whieh ooeur In the fotvola lor «, $*f(a) 4 //to) 4a,
has a geometrical meaning : a U the aagto ben reea the « axto aad the peipaiiSMlar
OX let fall upon the variable line from the orifhi: /{a) to the ittotaare OS tmm lie
origin to the variable line; and /'(<t) to, eJBBCpt for riga, the 4toiaawe JfJf itmm
the point }f where the variable line tooebei Its envelope to the Inol Jfof the
(liiMilar let fall upon the line from the origin. The feramhi lor • to effeei
Legendrt's formula.
432
PLANE CURVES
[X,§204
Fio. 39
804. Snrelope of a circle. Let us consider the family of circles
(16) (x-a)a+(y-6)2-p^ = 0,
where a, 6, and p are functions of a variable parameter t. The points where a
circle of this family touches the envelope are the points of intersection of the
circle and the straight line
(16) (X - a) a' + (y - b) 6' + pp' = 0.
This straight line is perpendicular to the Ungent MT to the curve C described
by the center (a, 6) of the variable circle (16), and its distance from the center is
p dp/ds, where s denotes the length of
the arc of the curve C measured from
some fixed point. Consequently, if the
line (16) meets the circle in the two
points N and N\ the chord NN' is
'c/ bisected by the tangent MT at right
angles. It follows that the envelope
y^ ^\ ^.^-'-yf^ I /' consists of two parts, which are, in
^"^ ^ ' general, branches of the same analytic
curve. Let us now consider several
special cases.
1) If /o is constant, the chord of con-
tact JVjV' reduces to the normal PP^ to
the curve C, and the envelope is com-
posed of the two parallel curves Ci and
a which are obtained by laying off the constant distance p along the normal,
on either side of the curve C.
2) If p = a + iT, we have p dp/ds — p, and the chord l^W reduces to the tan-
gent to the circle at the point Q. The two portions of the envelope are merged
into a single curve r, whose normals are tangents to the curve C. The curve C
is called the enolnte, of r, and, conversely, r is called an involute of C (see § 206).
If dp>d8j the straight line (16) no longer cuts the circle, and the envelope is
imaginary.
8ec(mdary caustics. Let us suppose that
the radius of the variable circle is propor-
tional to the distance from the center to a
fixed point O. Taking the fixed point as
the origin of coordinates, the equation of the
circle becomes
(« - a)* + {y- 6)« = lfl{a* + fta),
where I: is a constant factor, and the equation
of the chord of contact is
(« - a)a' + (y - b)l/ + k*{aa' + WO = 0.
If 8 and S' denote the distances from the
center of tlie circle to the chord of contact and to the parallel to it through the
origin, rcipectlvely, the preceding equation shows that d = k^S'. Let P be a
point on the radius MO (Fig. 40), such that MP = k^MO, and let C be the
Fig. 40
X»53W] CURVATURE
U$
locus of the center. Then the eqiuukm Joit f*« iiH ^mwi thtt IW flteid el qm
tact U the perpendionUr let lall from P upon the tenieiii to C ftlihe mmim M
Let ua suppoee tb*t ic in le« than unity, &nd let K denote that bcBneh of the
envelope which Hee on the lame tide of the t«nfent MT ee doee Um potet a
Let i and r, respectively, denote the two angles which the two Ums MO Md
liN make with the normal JT/ to the corre C. Then w «if| | ^f^
8ini = ^. rinr = i^. !*?^' = :?!>« IS „ »
Now let U8 imagine that the point O Ii a eooroe of light, and that the
■eparates a certain homogeneoue medium in which O lies from ai _
whose index of refraction with respect to the first is \/k. At\mt r«frac«kMi the
incident ray OM will be turned Into a refracted ray MR^ which, hy the law of
refraction, is the extension of the line NM. Hence all the ivfracted rays Jlffi
are normal to the envelope, which is called the mcondary eamMte ai rafraecloa.
The true caustic, that is, the envelope of the refracted raja, te the avotale of Ika
secondary caustic.
The second branch E' of the envelope evidently has no pbyileal
it would correspond to a negative index of refraction. If we set ft « 1,
envelope E reduces to the single point O, while tlie portion JT hofimw the loew
of the points situated symmetrically with with respect to the taofniu to C.
This portion of the envelope is also the secondary caustic qf r^ftedfam for Ind-
dent rays reflected from C which issue from the fixed pobit O. It may he shows
in a manner similar to the above that if a circle he described aboat each poiatsi
C with a radius proportional to the distance from its center to a isid slcri^tt
line, the envelope of the family will be a seoondary canstie with reipsel to a
system of parallel rays.
n. CURVATURE
205. Radius of curvature. The first idea of oairatare it that Um
curvature of one curve is greater than that of another if it
more rapidly from its tangent In order to render thia
vague idea precise, let us first consider the caae of a eirele. Its
curvature increases as its radius diminishes; ii is therefore quilt
natural to select as the measure of itt curratore the timplett fvae-
tion of the radius which increatet at the radint diminiahet, that
is, the reciprocal 1/R of the radiut. Let i4i} be an are of a eirek
of radius R which subtends an angle m at the oentar. The aafla
between the tangents at the extremities of the are ^B is alto«, aod
the length of the arc is « = Ru. Hence the measure of the earra-
ture of the circle is w/a. This last definition maj be esteadsd lo
an arc of any curve. Let .4 B be an aro of a plane eurre without a
point of inflection, and •* the angle between the tangent* at tha
extremities of the arc, the directions of the tansenta baiiif talWi
in the same sense according to some rule, — tha diftelioB froHi A
y
r
Bf
^
/j.
-^J^
^
— :?^
X
434 PLANE CURVES [X,§206
toward J5, for instance. Then the quotient <u/arc AB\^ called the
average Curvature of the arc AB. As the point B approaches the
point A this quotient in general approaches a limit, which is called
the curvature at the point A. The
radius of curvature at the point ^ is
defined to be the radius of the circle
which would have the same curvature
which the given curve has at the point
A\ it is therefore equal to the recipro-
cal of the curvature. Let s be the
length of the arc of the given curve
measured from some fixed point, and
^°' a the angle between the tangent and
some fixed direction, — the a? axis, for example. Then it is clear
that the average curvature of the arc ^i5 is equal to the absolute
value of the quotient Aa/As ; hence the radius of curvatui'e is given
by the formula
R = ± lim — = ± -r--
Acr da
Let us suppose the equation of the given curve to be solved for y
in the form y =f(x). Then we shall have
a = arc tan y\ da = -r^ -i ds = Vl -\- y'^dx,
and hence
(17) ie=±ii±p!.
Since the radius of curvature is essentially positive, the sign ±
indicates that we are to take the absolute value of the expression
on the right. If a length equal to the radius of curvature be laid
off from A upon the normal to the given curve on the side toward
which the curve is concave, the extremity / is called the center of
curvature. The circle described about / as center with R as radius
is called the circle of curvature. The coordinates (xq,. y^ of the
center of curvature satisfy the two equations
(1 + yT
which express the fact that the point lies on the normal at a dis-
taDce R from A. From these equations we find, on eliminating x^
(Xj - x) + {y, - y)y' = 0, (x, - xy + {y, - yY
X,5306] CURVATURE 435
In order to tell which sign should be taken, lei 111 note tluit if /* is
positive, as in Fig. 41, y^-y must be positive; henee the potttiTe
sign should be taken in this case. If y" is negatiTe, yi — y is neO'
tive, and the positive sign should be taken in this oaae ^h wi TIm
coordinates of the (>eni(>r of curvatare are therefore given bv tbi
formulae
(18) y.-. = ^'. x.-.-yi±J!lV
When the co5rdinate8 of a point (x, y) of the variable curve ate
given as functions of a variable parameter t^ we have, hy f 3S,
'' dx ^ da*
and the formulae (17) and (18) become
(19)
dx d*y — dy rf*x
dxd'y-dyd^x ^* ^ dxd^y-dyd^x
At a point of inflection y" = 0, and the radius of eorvatars b
infinite. At a cusp of the flrst kind y can be developed aeooiding
to powers of x^'^ in a series which begins with a term in x ; kaoea
y' has a finite value, but y" is infinite, and therefore the radios of
curvature is zero.
Noit. When the coordinates are expreved m foaetkas of Um are t of iIm
curve,
x = ^(»). If = f(»).
the functions and f satisfy the relation
0'«(s) + ^«(«) = l.
since dx^ \ d\f^ - dt^, and hence they also satisfy the rtfaUloo
^V + rf = 0.
Solving these equations for ^* and f , wt find
where « = ± 1, and the formula for the radius of owtataft tak« 00 U»»
cially elegant form
(20) ^= [«•)!• +[r(«)r.
436 PLANE CURVES [X,§206
206. Evolutes. The center of curvature at any point is the limit-
ing position of the point of intersection of the normal at that point
with a second normal which approaches the first one as its limiting
position. For the equation of the normal is
where X and Y are the running coordinates. In order to find the
limiting position of the point of intersection of this normal with
another which approaches it, we must solve this equation simulta-
neously with the equation obtained by equating the derivative of the
left-hand side with respect to the variable parameter x, i.e.
The value of F found from this equation is precisely the ordinate
of the center of curvature, which proves the proposition. It follows
that the locus of the center of curvature is the envelope of the
normals of the given curve, i.e. its evolute.
Before entering upon a more precise discussion of the relations
between a given curve and its evolute, we shall explain certain con-
ventions. Counting the length of the arc of the given curve in a
definite sense from a fixed point as origin, and denoting by a the
angle between the positive direction of the x axis and the direction
of the tangent which corresponds to increasing values of the arc,
we shall have tan a =±y', and therefore
,1 .dx
cos a = ± , — = ± -!-•
Vl -f y'^ ds
On the right the sign -f- should be taken, for if x and s increase
simultaneously, the angle a is acute, whereas if one of the varia-
bles X and 8 increases as the other decreases, the angle a is obtuse
($ 81). Likewise, if p denote the angle between the y axis and the
tangent, cos p = dy/ds. The two formulae may then be written
dx . dy
C08a = -;-> sina = -p->
ds ds
where the angle a is counted as in Trigonometry.
On the other hand, if there be no point of inflection upon the
given arc, the positive sense on the curve may be chosen in such a
way that 8 and a increase simultaneously, in which case R = ds/da
all along the arc. Then it is easily seen by examining the two
possible cases in an atrtual figure that the direction of the segment
X,fa08] CURV ATI-RE
4S7
starting at the point of the curve and going to the ocnter of etam.
ture makes an angle ai = a-^ ir/2 with the x axis. The ^>i.i> H f|n|„
(^i, yi) of the center of curvature are therefore given by th» ibnmOs
x, = x-h R coe(a' -f ^j = X - /? ain a,
yi = y-H/« sin ^a-f I) «y + iJooea,
whence we find
dxi = coaads — Rco8 a da— Bin a dR^-~ gin a dJt,
di/i=s Bin ads — R sin a da -^ COB a dH» oOBadR,
In the first place, these formulae show that dy^/dx^ ^ — efAa, which
proves that the tangent to the evolute is the normal to the giveo
curve, as we have already seen. Moreover
or rf«i = ± dR. Let us suppose for definiteness that the radius
of curvature constantly increases as we proceed along the cnrre C
(Fig. 42) from 3/^ to 3/^, and let us choose the poaitiTii mmm of
motion upon the evolute (/)) in such a way
that the arc ^i of (Z)) increases simultane-
ously with R. Then the preceding formula
becomes ds^ = dRy whence *i = i2 + C. It
follows that the arc /i l^ — R^ — /?i, and we
see that the length of any are of the evoluts
is equal' to the difference between the two
radii of curvature of the curve C whiek eoT'
respond to the extremities of that arc.
This property enables us to construct the
involute C mechanically if the evolute (D) be
given. If a string be attached to (D) at an
arbitrary point A and rolled around (£>) to /«, thence following the
tangent to A/,, the point ^ft will deacribe the inTolnto C at tiie
string, now held taut, is wound further on roond (/>). This ooo*
struction may be stated as follows : On each of the tangents iU of
the evolute lay off a distance IM = I, where / + « » oontt, « being
the length of the arc /< / of the evolute. Avigning Tarioot valnee
to the constant in question, an infinite number of involutes may be
drawn, all of which are obtainable from any one of Umb by lajing
off constant lengths along the normals.
438
PLANE CURVES
[X, §207
All of these properties may be deduced from the general formula
for the differential of the length of a straight line segment (§ 82)
dl = — da-i cos (Oj — dcr^ COS <i>2 .
If the segment is tangent to the curve described by one of its
extremities and normal to that described by the other, we may set
,oi = TT, ci>2 = 7r/2, and the formula becomes dl — da-i = 0. If the
straight line is normal to one of the two curves and I is constant,
dl = 0, cos Oil = Oj and therefore cos wj = 0.
The theorem stated above regarding the arc of the evolute depends
essentially upon the assumption that the radius of curvature con-
stantly increases (or decreases) along the whole arc considered. If
this condition is not satisfied, the statement of the theorem must
be altered. In the first place, if the radius of curvature is a maxi-
mum or a minimum at any point, dR = at that point, and hence
dxi = dyi = 0. Such a point is a cusp on
the evolute. If, for example, the radius
of curvature is a maximum at the point M
(Fig. 43), we shall have
arc//i = /JW-/iMi,
arc 7/2 = IM — I2M2J
whence
arc /1//2 = 2IM — I^M^ — I^M^.
Hence the difference I^M^ — I^M^ is equal
to the difference between the two arcs II^ and II^ and not their sum.
207. Cycloid. The cycloid is the path of a point upon the circumference of a
circle which rolls without slipping on a fixed straight line. Let us take the
fixed line m the x axis and locate the origin at a point where the point chosen on
the circle lies in the x axis. When the circle has rolled to the point / (Fig. 44)
the point on the circumference which was at O has come into the position Af,
X. $ 207] CURVATURE 4i9
where the circular arc /If i« equal to the n^imiiu OJ, Lai i
between the raUii CM aud CI an the rariable parameier. Tbao Iba
of the poiut if are
x= 0/-/P = H^-Baln^, v = J>1^ ^ IC -^ CQ^ R - Hvm^,
where P and Q are the proJeoUona of if on the two Uaca 01 and /T,
tively. It is easy to show that these fonnulai hold for any ralaa of tiM aofk #w
In one complete revolation tlie point whose path Is soocbt
OBOi . If the motion be continued indeflniiely, we obtain an
of arcs congruent to this one. From the praoedlng formuhe we find
X = ii(0 - sin 0) , dx = R{\-cM^)d4, iPzs R^n^d4^,
y = £(l-cos^), dy = Ruin^d^, #lf s Aoos^df*,
and the slope of the tangent is seen to be
dy tin^ ,»
— - = — = cot-»
dz l--cos^ a
which shows that the tangent at if la the ttntlght line if T, since tbe angk
MTC = <p/2, the triangle if rc being UMisceles. Hence the normal at if is tW
straight line MI through the point of tangency / of the fixed straight line with
the moving circle. For the length of the arc of the eyclold we find
ds'i = Ii^d<p^[s\n^<p-i-{\-coB^)*]=iR*ain*^d4^ or dssil{aia>4#.
if the arc be counted in the sense in which It increaaea with ^
the arc from the point as origin, we shall have
=4ii(i-co.|y
Setting = 2sr, we find that the length of one whole aeetton OBOi la tR. Tha
length of the arc 0MB from the origin to the m a ximnm B la tharafore 4R, aad
the length of the arc BM (Fig. 44) is 4R coa#/2. From the triaagia if TX7 Iha
length of the segment if T is 2R coe^/2 ; hence arc Bif = 2if T.
Again, the area up to the ordinate throng if Is
A =f*ydx =/*R»0 - «ooa^ + oot^f)d^
2coa^ +
/3 « t . ain2#\
= (-0-2sin^ + -^)
Hence the area bounded by the whole arc OBQx sii'i the base OOi » a««% ins4
is, three times the area qf the genaraUng eircU, (GAUtao.) ^^^
The formula for the radius of cunrature of a plana tnrm glvw lor the cycMd
8/J«sin«^d#«
o= ^ — = 4Raln?.
aJPsln«|d^
440
PLANE CURVES
[X, § 208
On the other hand, from the triangle MCI, MI = 2R sin 0/2. Hence p = 2MI,
and the center of curvature may be found by extending the straight line Ml
past / by its own length. This fact enables us to determine the evolute easily.
For, consider the circle which is symmetrical to the generating circle with
respect to the point I. Then the point M' where the line MI cuts this second
circle is evidently the center of curvature, since M'l = MI. But we have
arc T'M' = tcR - arc IM' = TtR - arc IM = xR - 01,
or
arc T'Jf' =OH-OI = IH= T'R.
Hence the point M' describes a cycloid which is congruent to the first one, the
cusp being at B' and the maximum at 0. As the point M describes the arc
BOi , the point 3f ' describes a second arc B'Oi which is symmetrical to the arc
OB" already described, with respect to BB".
208. Catenary. The catenary is the plane curve whose equation with respect
to a suitably chosen set of rectangular axes is
(21) y = |^e5 + e-5V
Its appearance is indicated by the arc MAM' in the figure (Fig. 45).
y
From (21) we find
If denote the angle which the tangent TM makes with the x axis, the formula
for / gives
Bin<f>
e" — e <*
1 Z'
COS0
e« -f- e <*
The ndios of curvature is given by the formula
r a'
But, Id the triangle MPN, MP = JlfJVcos ; hence
COB0 (t
X. $ 209] CURV ATURB 44 1
It followH that the radim of eiinratar« of the eummry !■ eqaal to iIm Imgtk •#
the normal MN. The etolnte may be found witboot dUBcolty fraa Uito *tf*
The length of the arc AM of the catenary la giran by tbo
■r=¥-''-i(-'--')
or a = V sin 0. If a perpendicular Fm ba dropped froBi i' ^i-ig. 4oj upoa i|m
tangent if 7, we tind, from the triangle Pifm,
jrm = Jn>aln^r=a.
Hence the arc ^Jf is equal to the dJitanco Mm.
209. Tractrix. The curve described by the point m (FIf. 4^ Is oUlsd lbs
tractrix. It is an involute of the catenary and has a eosp at lbs point A, Tbs
length of the tangent to the tractrix is the distance mP. But, in the iftaafis
MPm, mP = y cos = a ; hence the length mP measorsd akwg tbs lasfMl to
the tractrix from the point of tangency to the x axJs is ronitint and sqaal to «.
The tractrix is the only curve which hss this property.
Moreover, in the triangle MTP, Mm x mT = d*. Hence tbs prodoet of tbs
radius of curvature and the normal is a constant for tbs tractiiz. Tbiipioponj
is shared, however, by an infinite number of other pUns cartes.
The coordinates (xi, yi) of the point m are given by tbs
e« - « •
Xi = x — soos^ = g — a ,
S* + f"«
So
yi = y-«8in^ = -^^ -,
ei + «"«
or, setting e'/o = tan 9/2, the equations of the tractrix ars
log(tan?),
(22) xi = ocoe^ + alogitan- 1, v. -aninJ.
As the parameter varies from ir/2 to ir, the point lyj . yii aracnoMi tarn mrv
Avin, approaching the x axis as asymptote. As # varies (roa «/S to A, Ibo
point (xi , vi) describes the arc Am'n\ qrmmstrloal to tbs flat witb rsspsei to
the y axis. The arcs Amn and Am'n' oorrespoiid, wqwcllfsty, to tbs arcs AM
and AM' of the catenary.
210. Intrinsic equation. Let us try to
curve when the radius of curvature R Is
U = 0(s). Let or be the angle betwoan
Rsz± ds/da, and therefore
A first integration givet
X*i^
a = a,± . ^
442 PLANE CURVES [X,§210
and two further integrations give x and y in the form
x = Xo+ / C08ad«, y = yo + r sinada.
The curves defined by these equations depend upon the three arbitrary con-
stants Xo, Voi and oq. But if we disregard the position of the curve and think
only of its form, we have in reality merely a single curve. For, if we first con-
sider the curve C defined by the equations
the general formulae may be written in the form
X = Xo + -^cos ao — FsinoTo,
y = yo + -2" sin aro + Fcosoro,
if the positive sign be taken. These last formulae define simply a transforma-
tion to a new set of axes. If the negative sign be selected, the curve obtained
is symmetrical to the curve C with respect to the X axis. A plane curve is
therefore completely determined, in so far as its form is concerned, if its radius
of curvature be known as a function of the arc. The equation R = <p{s) is
called the inirinsic equation of the curve. More generally, if a relation between
any two of the quantities JS, s, and a be given, the curve is completely deter-
mined in form, and the expressions for the coordinates of any point upon it
may be obtained by simple quadratures.
For example, if R be known as a function of a, R=f{a)^ we first find
d» = f{a) da, and then
dx = cosa/(a:)da,
dy= sin. a f (a) da f
whence x and y may be found by quadratures. If i^ is a constant, for instance,
these formulas give
X = Xo + iJ sin a , y = yo — R cos a ,
and the required curve is a circle of radius R. This result is otherwise evident
from the consideration of the evolute of the required curve, which must reduce
to a single point, since the length of its arc is identically zero.
As another example let us try to find a plane curve whose radius of curva-
ture is proportional to the reciprocal of the arc, R = ays. The formulae give
and then
JC'ads ««
' — = — »
Although these integrals cannot be evaluated in explicit form, it ia easy to gain
an idea of the appearance of the curve. As s increases from to + oo, x and y
each pass through an infinite number of maxima and minima, and they approach
the same finite limit. Hence the curve has a spiral form and approaches
asymptotically a certain point on the line y = x.
X,52n] CONTACT OSCULATION
m. CONTACT OF PLANE CURVES
211. Order of conuct. Let C and C be two plana eurrct whkb
are tangent at some point A. To every point m on C lei ua uptifn,
according to any arbitrary law whatever, a point m' on C, the only
requirement being that the point m'
should approach A with m. Taking
the arc A in — or, what amounts to
the same thing, the chord Am — as
the principal infinitesimal, let us first
investigate what law of correspond-
ence will make the order of tlie infin-
itesimal 7n7n' with respect to i4m as
large as possible. Let the two curves
be referred to a system of rectangular
or oblique cartesian coordinates, the axis o/jfnai being jMralUi to tks
common tangent A T. Let
(C) >' = n*)
be the equations of the two curves, respectively, and let (j^, m) be
the coordinates of the point A, Then the coordinates of m will
be [a^o + K /(^o + A)], and those of m' will be [ac^ -h A, F{x^ -»- Ir)],
where k is a function of A which defines the ooirespondence be i n-eeo
the two curves and which approaches zero with h.
The principal infinitesimal Am may be replaced by A S3 ap^ ioit
the ratio 'a/>/.l7/i approaches a finite limit different from sero as the
point m approaches the point A, Let us now suppose that mm' is
an infinitesimal of order r -f 1 w ith r espect to A, for a certain
method of correspondence. Then mw? is of 0fd«r 2r 4- 2. If #
denote the angle between the axes, we shall have
^;^'' = [F(2-„ + A:) -fix. + A) + (ife - A) coarp + (* - A)«sin«#;
hence each of the differences k — k and F{x^ + k) —/{x^ + A) mnst
be an infinitesimal of order not less than r + 1, that ia,
;fc = A H- aA"-**, F(x. + *) -/t't + *) - fi^^^'f
where a and p are functions of A which approach finite limits as A
approaches zero. The second of these formolm may be written in
the form
F(xo + A -f aA"^») -yt'. + *) « ^A'**.
444 PLANE CURVES [X,§2ii
If the expression F(xq-{- h + ah"-*-^) be developed in powers of a,
the terms which contain a form an infinitesimal of order not less
than r -h 1. Hence the difference
A = F(x, + A) -f(x, 4- h)
is an infinitesimal whose order is not less than r-\-l and may exceed
r -f 1. But this difference A is equal to the distance mn between
the two points in which the curves C and C are cut by a parallel
to the 1/ axis through m. Since the order of the infinitesimal mm'
is increased or else unaltered by replacing m' by n, it follows that
the distance between two corresponding points on the two curves is an
infinitesimal of the greatest possible order if the two corresponding
points always lie on a parallel to the y axis. If this greatest possi-
ble order is r -f- 1, the two curves are said to have contact of order r
at the point A.
Notes. This definition gives rise to several remarks. The y axis
was any line whatever not parallel to the tangent A T. Hence, in
oi-der to find the order of contact, corresponding points on the two
curves may be defined to be those in which the curves are cut by
lines parallel to any fixed line D which is not parallel to the tan-
gent at their common point. The preceding argument shows that
the order of the infinitesimal obtained is independent of the direc-
tion of /), — a conclusion which is easily verified. Let mn and mm^
be any two lines through a point m of the curve C which are not
parallel to the common tangent (Fig. 46). Then, from the triangle
fH7n, fif
mm
mn sin mm' n
As the point m approaches the point A, the angles mnm' and mm'n
approach limits neither of which is zero or tt, since the chord m'n
approaches the tangent AT. Hence mm'/mn approaches a finite
limit different from zero, and mm' is an infinitesimal of the same
order as mn. The same reasoning shows that mm' cannot be of
higher order than mn, no matter what construction of this kind is
used to determine m' from 7n, for the numerator sin m7im' always
approaches a finite limit different from zero.
The principal infinitesimal used above was the arc ^m or the
chord Am. We should obtain the same results by taking the arc
An of the curve C for the principal infinitesimal, since Ani and An
are infinitesimals of the same order.
X,§212] CONTACT OSCULATION 446
If two curves C and C have a contaet of order r, the poioU m*
on C may be assigned to the pointe m on C in an infinite ntinikMr
of ways which will make mm' an infiniteaimai of order r 4- 1, — fur
that purpose it is sufficient to tet k ^ h-^ ah'**, where #^ r nod
where a is a function of A which remains finite for A b 0. On the
other hand, if « < r, the order of mm' cannot exceed « 4- 1.
212. Analytic method. It follows from the preceding seeiioo thai
the order of contact of two curves C and C* is given by evaluating
the order of the infinitesimal
y-y = F(x, + A)--y(T, + A)
with respect to h. Since the two curves are tangent at A,
/\xo) =f(xo) and F'(xo) =/'(3?o)- It may happen that others of the
derivatives are equal at the same point, and we shall suppose for
the sake of generality that this is true of the first n derivatives :
r>3^ \ F(x„)=f{x,), FXx,)^r(',),
but that the next derivatives f '•♦"(!,) and /••♦ "(i,) aw nnaqaaL
Applying Taylor's series to each of the functions F(x) aad/{xf, w«
find
or, subtracting,
(24) >^-y = i.2.^-(l-n) ^^'*"^'^^--^'"''^'^^'"*^'
where c and e' are infinitesimals. H/oUowm tAmi tMe^fdermfi
of two ctirves is equal to the orrfw n 0/ ths higheti derivaii9t» ^f Fi^z)
and fix) which are equal for x = x,.
The conditions (23), which are due to Lagrange, are the neeeesaiy
and sufficient conditions that ac « x^ should be a multiple root of
order n -f 1 of the equation F(x) =/(x). But the rooto of thit
equation are the abecissn of tho iminu of intersection of the two
446 PLANE CURVES [X, §212
curves C and C ; hence it may be said that two curves which have
contact of order n have n -{• 1 coincident points of intersection.
The equation (24) shows that Y — y changes sign with A if ti is
even, and that it does not if n is odd. Hence curves which have
contact of odd order do not cross, but curves which have contact of
even order do cross at their point of tangency. It is easy to see why
this should be true. Let us consider for definiteness a curve C'
which cuts another curve C in three points near the point A. If
the curve C" be deformed continuously in such a way that each of
the three points of intersection approaches A, the limiting position
of C has contact of the second order with C, and a figure shows that
the two curves cross at the point A. This argument is evidently
generaL
If the equations of the two curves are not solved with respect to
Y and y, which is the case in general, the ordinary rules for the
calculation of the derivatives in question enable us to write down
the necessary conditions that the curves should have contact of
order n. The problem is therefore free from any particular diffi-
culties. We shall examine only a few special cases which arise
frequently. First let us suppose that the equations of each of the
curves are given in terms of an auxiliary variable
( x=f(t), iX=^f(u),
(C) \ -^^^^ (C) \ -^^ ^'
and that \f/(to) = </>(^o) ^^^ ^'(^o) — <^'(^o)j i-^- t^3,t the curves are tan-
gent at a point A whose coordinates are f(t(i), <t>(to). Iff'{t^ is not
zero, as we shall suppose, the common tangent is not parallel to the
y axis, and we may obtain the points of the two curves which have
the same abscissae by setting u = t. On the other hand, x — Xq\q of
the first order with respect to ^ — ^o> and we are led to evaluate the
order of i/^(^) — <^(^) with respect tot — tQ, In order that the two
curves have at least contact of order n, it is necessary and sufficient
that we should have
(26) ^^,{t,) = <A(^o) , •A'(^o) = <i>\to), • • • , lA^"^ (« = <^^"^ (^o) ,
and the order of contact will not exceed n if the next derivatives
^•■^'>(^) and <^<'' + '>(^o) are unequal.
Again, consider the case where the curve C is represented by the
two equations
(26) a^'/lO. y = *(0.
X, i 212]
CONTACT OSCULATION
447
and the curve C by the single equatioD F(x, y) ■■ 0. This cam nuiy
be reduced to the preceding by replacing » in F(m^ y) by /(I) and
considering the implicit function y a ^(t) defined by the equaltoo
(27) F[/(<),f(0]-0.
Then the curve C is also represented by two eqoaliona of Um
(28)*
*=/(0, y = ^'V
In order that the curves C and C should have contact ol Older m at
a point A which corresponds to a value t^ of the parameter, it is
necessary that the conditions (25) should be satisfied. But the
successive derivatives of the implicit function ^t) are giTeo by the
equations
(29)
d"F
^[A0/+ +X7^-'W
Hence necessary conditions for contact of order n will be
by inserting in these equations the relations
f = f„ X =/(fo), ^*o) = ^(^o), f (« = ♦'(<•)» • • •» r'\t.) = ♦-'(!•).
The resulting conditions may be expressed as foUowa :
Let
F(o=nAO» ♦{')];
then the two given curves teill have <U Utui eamiati ^ 0rdw if mmd
only if
(30) F(fo) = 0, F'(<.) = 0, p*»{g-o.
The roots of the equation f{t) » are the valnee of I which cot^
respond to points of intersection of the two giTcn oonrea. Heaee
the preceding conditions amount to saying thai « i- ^ ia a mvltiple
root of order n, i.e. that the two ourree have a + 1
of intersection.
448 PLANE CURVES [X,§213
213. Osculating curves. Given a fixed curve C and another cm-ve
C which depends upon n + 1 parameters ay bf Cj •• -, l,
(31) F(Xyy, a, b,cy-,r) = 0,
it is possible in general to choose these n -{- 1 parameters in such a
way that C and C shall have contact of order n at any preassigned
point of C. For, let C be given by the equations x =f(t), y = <f>(t).
Then the conditions that the curves C and C should have contact
of order n at the point where t = tQ are given by the equations (30),
where
^(t) = F[f(t),<f>(t),a,b,c,--',l^.
If Iq be given, these n -fl equations determine in general the n+l
parameters a, b, c, •••, /. The curve C obtained in this way is
called an osculating curve to the curve C.
Let us apply this theory to the simpler classes of curves. The
equation of a straight line y = ax -{- b depends upon the two param-
eters a and b ; the corresponding osculating straight lines will have
contact of the first order. If y =f{x) is the equation of the curve C,
the parameters a and b must satisfy the two equations
f(xo) = aXo-hb, f(x,) = a',
hence the osculating line is the ordinary tangent, as we should
expect.
The equation of a circle
(32) {x - ay +(y-by-R^ =
depends upon the three parameters a, b, and R ; hence the corre-
sponding osculating circles will have contact of the second order.
Let y =f(x) be the equation of the given curve C ; we shall obtain
the correct values of a, b, and R by requiring that the circle should
meet this curve in three coincident points. This gives, besides the
equation (32), the two equations
(33) x-a + (y- b)y' = 0, 1 + y^ + (y _ b)y" = 0.
The values of a and b found from the equations (33) are precisely
the coordinates of the center of curvature (§ 206) ; hence the oscu-
lating circle coincides with the circle of curvature. Since the con-
tact is in general of order two, we may conclude that in general the
circle of curvature of a plane curve crosses the curve at their point
of tangency.
X.f213] CONTACT OSCULATION 449
All the above results might hare been foraum • fH&ri, For,
since the coordinates of the center of cumlUM dnMad miIt ^
X, y, y'y and y", any two curves which have oooteet of IIm f^n td
order have the same center of curvature. But the eeoter of tur^
ture of the osculating circle is evidently the center of thai eirato
itself; hence the circle of curvature most eoinotde witk tlie oms*
lating circle. On the other hand, let us consider two circl« of
curvature near each other. The difference between their radii,
which is equal to the arc of the evolute between the two <*^tttiftw,
is greater than tlie distance between the centers; hence one of
the two circles must lie wholly inside the other, which eoold not
happen if both of them lay wholly on one side of the eurra C in
the neighborhood of the point of contact. It follows that tbay
cross the curve C
There are, however, on any plane curve, in general, certain pointa
at which the osculating circle does not cross the curve; this ezcep*
tion to the rule is, in fact, typical. Given a curve C which depoada
upon n + 1 parameters, we may add to the n + 1 equations (SO) tba
new equation
provided that we regard t^ as one of the unknown qnantitiea
determine it at the same time that we determine the
ay b, Cj 'y I. It follows that there are, in general, on any plana
curve C, a certain number of points at which the order of oon-
tact with the osculating curve C is it + 1. For example, there ara
usually points at which the tangent has contact of the aeoood order;
these are the points of inflection, for which y" « 0. In ordar to find
the points at which the osculating circle haa oontaot of the tliird
order, the last of equations (33) must be differentiated again, whieli
gives
3y'y" + (y-*)y"' = 0,
or finally, eliminating y — by
(34) (l + y'«)y"'-3yy = 0.
The points which satisfy this last condition are thoaa for which
dR/dx = 0, i.e. those at which the radius of eorvatora ia a maxi-
mum or a minimum. On the ellipse, for example, tbeae poinU are
the vertices ; on the cycloid they are the poinU at which the tan-
gent is parallel to the base.
450 PLANE CURVES [X,§214
214. Osculating curves as limiting: curves. It is evident that an
osculating curve may be thought of as the limiting position of a
curve C which meets the fixed curve C in n-^1 points near a fixed
point A of C, which is the limiting position of each of the points
of intersection. Let us consider for definiteness a family of
curves which depends upon three parameters a, b, and c, and let
^0 + ^1 » ^0 + ^a> a.nd to + As be three values of t near t^. The curve
C which meets the curve C in the three corresponding points is
given by the three equations
(35) F(^o + Ai) = , F(^o 4- ^2) = , F(^o + ^3) = .
Subtracting the first of these equations from each of the- others and
applying the law of the mean to each of the differences obtained,
we find the equivalent system
(36) F(fo + ^) = , F'(^o -\-h) = 0, F(t, + A:^) = ,
where ki lies between hi and Agj and kz between hi and h^. Again,
subtracting the second of these equations from the third and apply-
ing the law of the mean, we find a third system equivalent to either
of the preceding,
(37) F(to-hhi) = 0, F'(^o + A;i) = 0, F"(^o + ^i)=0,
where Zi lies between ki and kz- As hi, h^, and h^ all approach
zero, ki, k^j and li also all approach zero, and the preceding equa-
tions become, in the limit,
F(^o) = 0, F'(^o) = 0, F"(^o) = 0,
which are the very equations which determine the osculating curve.
The same argument applies for any number of parameters whatever.
Indeed, we might define the osculating curve to be the limiting
position of a curve C which is tangent to C at jo points and cuts C
at q other points, where 2p + g' = n + 1, as all these p -\- q points
approach coincidence.
For instance, the osculating circle is the limiting position of a
circle which cuts the given curve C in three neighboring points. It
is also the limiting position of a circle which is tangent to C and
which cuts C at another point whose distance from the point of
tangency is infinitesimal. Let us consider for a moment the latter
property, which is easily verified.
Let us take the given point on C as the origin, the tangent at
that point as the x axis, and the direction of the normal toward the
X,Ex..]
EXERCI8E8
461
center of curvature as the positiTe direetion of tli« y axi«. At tiM
origin, y' = 0. lience R = 1/y", and theraforo, by Taylor't werim.
y^^'fe-*-*)'
where c approaches zero with x. It fol-
lows that It is the limit of the expres-
sion xV(2y) = 0P*/(2MP) at the point
M approaches the origin. On the other
hand, let Ki be the radius of the circle
Ci which is tangent to the 2; axis at the
origin and which passes through M,
Then we shall have
rio.«7
OP* = Mm^ = MP(2Rt - MP),
or
OP*
2AfP
MP
hence the limit of the radius 7?, is really equal tn Um radius of
curvature R
1. Apply the general formul» to find the eTolale of sa elllpM : of sa hyper-
bola ; of a parabola.
2. Show that the radius of cnrvsture of a conic Is pct^iorUoiuU lo lbs 4whs
of the segment of the normal between its points of intsissdioii with tht ew<t
and with an axis of symmetry.
3. Show that the radius of curvature of the psrshohi is eqoal to twiet ihe
segment of the normal between the curve and the dlrselriz.
4. Let F and i?" be the foci of an eUipee, M s point oa tbo sUipss, MN Ihe
normal at that point, and N the point of intersection of that aorauU sad the
major axis of the ellipse. Erect s perpendicnlsr NK to MN st N, aiesfinf MF
at K. At K erect a perpendicular KO to MF, meeting MN st O.
O is the center of curvature of the ellipse at the point M,
6. For the extremities of the major axis the
illusory. Let A OA' be the uMJor axis and BO^B the niaor axis of the
On the segments OA and OB construct the rectaagle OA KB Frtm JT Ist tell
a perpendicular on A B, meeting the major and minor axes st C snd iX raipeo*
tively. Show that C snd D are the centers of curratare of the sUipas for llM
pointj) A and B, respectively.
6. Show that the evolute of the spiral p s mr^ Is a
given spiral.
toihs
462 PLANE CURVES [X, Em.
7. The path of any point on the circumference of a circle which rolls with-
out slipping along another (fixed) circle is called an epicycloid or an hypocycloid.
Show that the e volute of any such curve is another curve of the same kind.
8. Let AB he an arc of a curve upon which there are no singular points and
no points of inflection. At each point m of this arc lay off from the point m
along the normal at m a given constant length I in each direction. Let mi and
m« be the extremities of these segments. As the point m describes the arc AB,
the points mi and ma will describe two corresponding arcs AiBi and ^2^2-
Derive the formulas Si = S — ld, 82 = 8 + 16, where S, Si, and S2 are the
lengths of the arcs AB, AiBi, and ^2 -^2 , respectively, and where 6 is the angle
between the normals at the points A and B. It is supposed that the arc -^i^i
lies on the same side of AB as the evolute, and that it does not meet the evolute.
[Licence, Paris, July, 1879.]
9. Determine a curve such that the radius of curvatures p at any point M
and the length of the arc s = AM measured from any fixed point A on the curve
satisfy the equation ds = p^ ■{■ a^, where a is a given constant length.
[Licence, Paris, July, 1883.]
10. Let C be a given curve of the third degree which has a double point
at O. A right angle MON revolves about the point O, meeting the curve C in
two variable points 3f and N. Determine the envelope of the straight line MN.
In particular, solve the problem for each of the curves Xy2 = x^ and x^-\-y^ = itxy.
[Licence, Bordeaux, July, 1885.]
11. Find the points at which the curve represented by the equations
X = a (nw — sin ft>) , y — a{;n — cos w)
has contact of higher order than the second with the osculating circle.
[Licence, Grenoble, July, 1885.]
12. Let m, mi , and m2 be three neighboring points on a plane curve. Find
the limit approached by the radius of the circle circumscribed about the triangle
formed by the tangents at these three points as the points approach coincidence.
13. If the evolute of a plane curve without points of inflection is a closed
curve, the total length of the evolute is equal to twice the difference between the
sura of the maximum radii of curvature and the sum of the minimum radii of
curvature of the given curve.
14. At each point of a curve lay off a constant segment at a constant angle
with the normal. Show that the locus of the extremity of this segment is a
curve whose normal passes through the center of curvature of the given curve.
16. Let r be the length of the radius vector from a fixed pole to any point of
a plane curve, and p the perpendicular distance from the pole to the tangent.
Derive the formula R = ± rdr/dp, where R is the radius of curvature.
16. Show that the locus of the foci of the parabolas which have contact of
the second order with a given curve at a fixed point is a circle.
17. Find the locus of the centers of the ellipses whose axes have a fixed direc-
tion, and which have contact of the second order at a fixed point with a given
curve.
CHAPTEK XI
8KSW CURVES
I. OSCULATING PLANE
215. Definition and equation. Let AfThe the tangent at a point M
of a given skew curve r. A plane through MT and a point i/' of
r near M in general approaches a limiting position aa the point !#'
approaches the point M. If it does, the limiting position of the
plane is called the osculating plane to the curre F at the point i#.
We shall proceed to find its equation.
Let
(1) ^=/(0. y = *(0» * = «0
be the equations of the curve T in terms of a parameter f, and let t
and t -\- hhQ the values of t which correspond to the pointa J# and
M\ respectively. Then the equation of the plane MTM* it
^(X - X) + B(r - y) + C(Z - «) = 0,
where the coefficients Aj B, and C muat satiafy the two relatiooa
(2) Af(t) + B4>\t)-\-Ci^(t)^0,
(3) /l[/(?-hA)-y];0]+^W + A)-^(0]+C[f(r + A;-^tOj-0.
Expanding f(t 4- h), ^{t + A) and ^(< -f A) by Tajlor'a aartea, tkm
equation (3) becomes
After multiplying by h, let us subtract from thia equation the eqaar
tion (2), and then divide both sides of the reaulttng eqaatkm by
A72. Doing 80, we find a system equivalent to (2) and (3):
Aif"(t) + «o + B[r(t) + •.] + nrw + -] - 0,
where c,, c„ and c, approach lero with A. In the limit aa k
approaches zero the second of theae eqoatioiia
(4) AfV) + ^♦"(0 + ^^W - ^
4AS
464 SKEW CURVES [XI,§215
Hence the equation of the osculating plane is
(6) A{X -x)-h B(Y-y) + C(Z-z) = 0,
where A, B, and C satisfy the relations
Adx -^Bdy -\- C dz =0,
^^^ \Ad^x + B d^y + Cd^z = 0.
The coefficients A, B, and C may be eliminated from (5) and (6),
and the equation of the osculating plane may be written in the form
X-x Y-y Z-z
dx dy dz
d^x d^y d^z
= 0.
Among the planes which pass through the tangent, the osculating
plane is the one which the curve lies nearest near the point of tan-
gency. To show this, let us consider any other plane through the
tangent, and let F{t) be the function obtained by substituting
f{t -f h), <f>{t + h), \J/(t + h) for X, Y, Z, respectively, in the left-hand
side of the equation (5), which we shall now assume to be the equa-
tion of the new tangent plane. Then we shall have
F(.t) = O f^-^"(*> + ^*"<^*) + ^"^"(') + ''^'
where rj approaches zero with h. The distance from any second
point M' of r near M to this plane is therefore an infinitesimal of
the second order; and, since F(t) has the same sign for all sufficiently
small values of h, it is clear that the given curve lies wholly on one
side of the tangent plane considered, near the point of tangency.
These results do not hold for the osculating plane, however. For
that plane, Af" + B<l>" + C\f/" = ; hence the expansions for the
coiirdinates of a point of T must be carried to terms of the third
order. Doing so, we find
^W- 17273 V dg^ '^V'
It follows that the distance from a point of V to the^ osculating
plane is an infinitesimal of the third order ; and, since F({) changes
sign with A, it is clear that a skew curve crosses its osculating plane
at their common point. These characteristics distinguish the oscu-
lating plane sharply from the other tangent planes.
XI, §216] 08CULATlN(i I'LANK 455
216. Stationary otcalfttiiig pkne. The r«ralU Jntt obtaiatd an Boi
valid if the coetiicieiits A^ Bf C of the oeculating pUoe mMkdw the
relation
(7) i4d««4-Brf«y + Crf««B0.
If this relation is satisfied, the expansions for the oo6rdinatee muii
be carried to terms of the fourth order, and we should obtain a
relation of the form
h< / Ad*x-^Bd*y^Cd*z \
^^^^ 1.2.3.4V dt' ^V'
The osculating plane is said to be stationary at any point of r for
which (7) is satisfied ; if A d^x + Bd^y 4- C d^m does sot vaiiiah
also, — and it does not in general, — F{t) changes sign with k and
the curve does not cross its osculating plane. Moreover the distanee
from a point on the curve to the osculating phuie at fuch a point ta
an infinitesimal of the fourth order. On the other hand, if the
relation A d*x -^ Bd*i/ -^ C d*x ^ is satisfied at the same point,
the expansions would have to be carried to terms of the fifth order ;
and so on.
(7). we
= 0,
whose roots are the values of t which oorrespond to the points of f
where the osculating plane is stationary. There are then, usually,
on any skew curve, points of this kind.
This leads us to inquire whether there are curves all of whose
osculating planes are stationary. To be precise, let us try to find
all the possible sets of three functions x, y, s of a single vaHahle #,
which, together with all their derivatives up to and including those
of the third order, are continuous, and which satisfy the equatioo
(8) for all values of t between two limits atmdk(a<b).
Let us suppose first that at least ooe of the minors of A which
correspond to the elements of the third row, wkydxd^y-d^ ^ jr, does
not vanish in the interval (a, h). The two equations
W ( ,1. = Ctd'x + C,J'g,
Eliminating A, By and C between the
obtain the equation
dx dy dx
(8) A =
d}x d^y d^x
d^x d^y </•«
456 SKEW CURVES [XI, §216
which are equivalent to (6), determine Ci and Cg as continuous
functions of t in the interval (a, b). Since A = 0, these functions
also satisfy the relation
(10) d^z = Cid^x + Cg d'^y.
Differentiating each of the equations (9) and making use of (10),
we find
dC.dx + dC^dy = 0, dC^d^x + dC^d^y = 0,
whence dCi = dC^ = 0. It follows that each of the coefficients Ci
and Ci is a constant; hence a single integration of the first of
equations (9) gives
z = CiX + C^y-\'C^y
where C, is another constant. This shows that the curve r is a
plane curve.
If the determinant dxd^y — dyd^x vanishes for some value c of the variable t
between a and 6, the preceding proof fails, for the coefficients Ci and d might
be infinite or indeterminate at such a point. Let us suppose for definiteness
that the preceding determinant vanishes for no other value of t in the interval
(a, 6), and that the analogous determinant dx(Pz — dzd^x does not vanish for
t = c. The argument given above shows that all the points of the curve r which
correspond to values of t between a and c lie in a plane P, and that all the
points of r which correspond to values of t between c and b also lie in some
plane Q. But dxd^z —dzd'^x does not vanish for t = c; hence a number h
can be found such that that minor does not vanish anywhere in the interval
(c — A, c ■{■ h). Hence all the points on T which correspond to values of t
between c — h and c -\- h must lie in some plane R. Since R must have an
infinite number of points in common with P and also with Q, it follows that
these three planes must coincide.
Similar reasoning shows that all the points of r lie in the same plane unless
all three of the determinants
dxd^y -dyd^x, dxd^z — dzd^x, dyd^z — dzd^y
vanish at the same point in the interval (a, h). If these three determinants do
vanish simultaneously, it may happen that the curve V is composed of several
portions which lie in different planes, the points of junction being points at
which the osculating plane is indeterminate.*
II all three of the preceding determinants vanish identically in a certain
intefTal, the curve r is a straight line, or is composed of several portions of
straight lines. If dx/dt does not vanish in the interval (a, 6), for example, we
may write
a^ydx -dyd^z _ ^ d^zdx -dzd^x _ ,
whence
dy = CidXy dz = Cadx,
•This singular case seems to have been noticed first by Peano. It is evidently of
Interest only from a purely analytical standpoint.
XI. §217] OSCULATING PLAHK 4^7
where Ci and Ct are oomtiiili. FbuOly, uoUmt »-1^trmim 0n§
which shows that r is a straight line.
S17. Stationary taag«ato. The pireoeding parifraph MgpaU tJ
certain points on a skew curve whieh we had not pi ti kMM|y
the points at which we hate
(11) *£ = *^ = *f.
^ ' dx dy dM
The tangent at such a point la eald to be iiatianarjf. it u eaaj to i
formula for the distance between a point and a ■M^fh< Una UmI Ite
from a point of r to the tangent at a neighboring point, whIeh it
infinitesimal of the second order, ia of the third order for a
If the given curve F is a plane cunref the itatiftnary Ungenti are tbt Unfiliel
the poinu of inflection. The preceding paragraph ihowi that tha only eortt
whose tangents are all stationary ia the itraight Una.
At a point where the ungent is stationary, A = 0, and the aqnatloa ol Iht
osculating plane becomes indeterminate. But in general thia ladolemlaatkM
can be removed. For, returning to the calcnlation at the heglnnlag of | SIS
and carrying the expansions of the oottrdinaiee of Jf ' to terms of tho third crter,
it is easy to show, by means of (11), that the equation of the pliM thro«g|l JT
and the tangent at 3f is of the form
X-x Y-y Z-t
nt) ^'(0 f(o
0.
where ci , cj , e* approach zero with k. Hence that plane appitMMM a ;
definite limiting position, and the equation of the oa cn l ating plane li
replacing the second of equations (0) by the aqoadon
^d»z + Bd»y + Cd"« = 0.
If the coordinates of the point M also satisfy ths sqantlOB
d»g _ d^W _d^*
dx ^ dy " dx'
the second of the equations (6) should be replsosd by ths sqnattos
^dfx = B<itif-f CdrsKO,
where q is the least integer for which this lattsr eqnatkm is dlMlMC tnm ths
equation Adx= Bdy + Cdx t= 0. Ths proof of this sti
nation of the behavior of the curre with rs^MOt to its os rohtin g plaas sfs Ml
to the reader.
Usually the preceding equaUon Inrolring the third diflerentials is «AflisM»
and the coefficienU ^, B, C do not satisfjr ths s^oslkm
ii d>s •»- Bdhy + C#t ai 0.
In this case the curre crosses s?sry tangent phuis sxespt ths
458 SKEW CURVES [XI, §218
218. Special earvet. Let us consider the skew curres F which satisfy a
relation of the form
(12) xdy -ydx = Kdz,
where IT is a given constant. From (12) we find immediately
( xcPy- yd^x = KcPz,
***' \xd*y-yd»x + dzd^y-dyd^x = Kd^z.
Let us try to find the osculating plane of T which passes through a given point
(a, 6, c) of space. The coordinates (x, y, z) of the point of tangency must satisfy
the equation
a — X b — y c — z
dx dy dz = 0,
d^x d^y d^z
which, by means of (12) and (13), may be written in the form
(14) ay-bx-\-K(c-z) = 0.
Hence the possible points of tangency are the points of intersection of the
curve r with the plane (14), which passes through (a, 6, c).
Again, replacing dz, d^z and d^z by their values from (12) and (13), the equa-
tion A = 0, which gives the points at which the osculating plane is stjationary,
becomes
A = ^{dxd^y -dyd^xY = 0;
hence we shall have at the same points
d^x _ d2y ^ yd^x — xd'^y _ d^
dx dy ydx — xdy dz
which shows that the tangent is stationary at any point at which the osculating
plane is stationary.
It is easy to write down the equations of skew curves which satisfy (12) ; for
example, the curves
x = At^, y=:Bt^, z = Ct«' + «,
where A, B, C, m, and n are any constants, are of that kind. Of these
the simplest are the skew cubic x = t, y = t^j z = t', and the skew quartic
x = t, y = t^,z = t^. The circular helix
05 = a cost, y = aamt, z = Kt
is another example of the same kind.
In order to find all the curves which satisfy (12), let us write that equation in
the form
d{xy - Kz) = 2ydx.
If we Mt
«=/(t), xy-Kz = <f>{t),
the preceding equation becomes
2y/'(0 = ^'(0.
XI, §219] ENVELOPES OF .mk^al^.jj 469
Solving these three equatioiui (or x, y, and s, w iad tiM fMMil tqmi^imm «C P
in the form
where /(t) and ^{t) are arbitrary fnnctiona of the paraaMler t It la d«r, llov-
ever, that one of these fanctloos may be iiffnml at laadom wIthoiH iam of
generality. In fact we may set /(f) = f, daoe this aaoanli to drnMlac/tr) aa a
new parameter.
n. ENVELOPES OF SURFACES
Before taking up the study of the ounrature of ikew ennret, w%
shall discuss the theory of enyelopes of surfaoea.
219. One-parameter families. Let 5 be a surface of the family
(16) /[x,y,«,a) = 0,
where a is the variable parameter. If there exiata a surfaoe B which
is tangent to each of the surfaces S along a curve C, the torfiee S
is called the envelope of the family (16), and the curre of
C of the two surfaces S and E is called the eharaeUritiie
In order to see whether an envelope exists it is evidently
sary to discover whether it is possible to find a curve C oo eaeh of
the surfaces S such that the locus of all these carves is taafeni to
each surface S along the corresponding curve C Let (x, y, u) be
the coordinates of a point A/ on a oharacteristio. If 1/ is ooi a
singular point of S^ the equation of the tangent plane to 5 al if it
As we pass from point to point of the surfaoe i?, «, jf , c, and a are
evidently functions of the two independent variables whieh express
the position of the point upon K, and theee functions satisfy the
equation (16). Hence their differentials satisfy the relation
Moreover the necessary and sufficient oonditioik that the
plane to E should coincide with the taogeol plane lo S is
or, by (17),
(18) g-0.
460 SKEW CURVES [XI, §220
Conrepsely, it is easy to show, as we did for plane curves (§ 201),
that the equation R(xj y, z) = 0, found by eliminating the param-
eter a between the two equations (16) and (18), represents one or
more analytically distinct surfaces, each of which is an envelope
of the surfaces S or else the locus of singular points of 5, or a com-
bination of the two. Finally, as in § 201, the characteristic curve
represented by the equations (16) and (18) for any given value of a
is the limiting position of the curve of intersection of S with a
neighboring surface of the same family.
220. Two-parameter families. Let S be any surface of the two-
parameter family
(19) f(x,y,z,a,b)=0,
where a and b are the variable parameters. There does not exist,
in general, any one surface which is tangent to each member of this
family all along a curve. Indeed, let b = <l>(a) be any arbitrarily
assigned relation between a and b which reduces the family (19) to
a one-parameter family. Then the equation (19), the equation
b = ^(a), and the equation
represent the envelope of this one-parameter family, or, for any
fixed value of a, they represent the characteristic on the correspond-
ing surface S. This characteristic depends, in general, on '<t>'(a),
and there are an infinite number of characteristics on each of the
surfaces S corresponding to various assignments of <^(a). There-
fore the totality of all the characteristics, as a and b both vary arbi-
trarily, does not, in general, form a surface. We shall now try to
discover whether there is a surface E which touches each of the
family (19) in one or more points, — not along a curve. If such a
surface exists, the coordinates («, y, z) of the point of tangency of
any surface S with this envelope E are functions of the two variable
parameters a and b which satisfy the equation (19) ; hence their dif-
ferentials rfx, rfy, dz with respect to the independent variables a
and b satisfy the relation
XI. §221] SNVELOPBS OF SURPACES 4«1
Moreover, in order that the surface which it Um loeui of tiM pofail
of tangency (or, y, z) should be tan^oat to 5, U it alto innmiij
that we should have
or, by (21),
Since a and b are independent variables, it follows that the equat¥nit
must be satisfied simultaneously by the coordinates (as, y, u) of tiM
point of tangency. Hence we shall obtain the eqiuKion of tho
envelope, if one exists, by eliminating a and b between the thftt
equations (19) and (22). The surface obtained will sorely be tan-
gent to ^' at (x, y^ z) unless the equationa
dx dy d*
are satisfied simultaneously by the values (x, y, «) whieh tatiafjr (19)
and (22) ; hence this surface is either the envelope or else the locos
of singular points of S.
We have seen that there are two kinds of envelopes, **tt**^
on the number of parameters in the given family. For example,
the tangent planes to a sphere form a two-paramotar family, and
each plane of the family touches the surface at only one point
On the other hand, the tangent planes to a oone or to a eylinder
form a one-parameter family, and each member of the lunily it
tangent to the surface along the whole length of a
221. Developable surfaces. The envelope of any one-parfttar JMnily
of planes is called a developabU tur/ace. Let
(23) « = aar + y/(a) + 4(a)
be the equation of a variable plane P, whtre a is a ^liimtlM nod
where /(a) and 4>(a) are any two fonetions of a, Tbtn Ihn 9qu^
tion (23) and the equation
(24) « + yA«) + ♦'(«)-<>
repi-esent the envelope of the family, or, for a givtn Tihitof «, Ibtj
represent the characteristic on the ooRttpooding plants All Ibitt
462 SKEW CURVES [XI, §221
two equations represent a straight line; hence each characteristic
is a straight line G, and the developable surface is a ruled surface.
We proceed to show that all the straight lines G are tangent to the
same skew curve. In order to do so let us differentiate (24) again
with regard to a. The equation obtained
(26) y/"(«) + ^"(a) =
determines a particular point M on G. We proceed to show that G
is tangent at M to the skew curve V which M describes as a varies.
The equations of T are precisely (23), (24), (25), from which, if we
desired, we might find jc, y, and z as functions of the variable
parameter a. Differentiating the first two of these and using the
third of them, we find the relations
(26) dz = adx -\-f{a)dy, dx +f{a) dy = 0,
which show that the tangent to T is parallel to G, But these two
straight lines also have a common point ; hence they coincide.
The osculating plane to the curve r is the plane P itself. To
prove this it is only necessary to show that the first and second
differentials of a, y, and z with respect to a satisfy the relations
dz = adx +f(cc) dy,
d:^z = ad^x-{-f(a)d^y.
The first of these is the first of equations (26), which is known to
hold. Differentiating it again with respect to or, we find
d^z = ad^x ^f(a)d^y + [dx +f(a)dy^da,
which, by the second of equations (26), reduces to the second of the
equations to be proved.
It follows that any developable surface may he defined as the locus
of the tangents to a certain skew curve T. In exceptional cases the
curve r may reduce to a point at a finite or at an infinite distance ;
then the surface is either a cone or a cylinder. This will happen
whenever f"(a) = 0.
Conversely, the locus of the tangents to any skew curve r is a
developable surface. For, let
be the equations of any skew curve T. The osculating planes
A(X - x) + B(Y -y)-hC(Z-z) =
XI. §221] ENVELOPES OF SURFACES
form a ono-parameter lamilj, whose eDT«lop« it giv«o bj Um pi»
cediug equation and the equation
rf^ ( X - «) + rf5( r - y) + dC(Z ^ .) - .
For any fixed value of t the same equations represent the ebarse-
teristic in the corresponding osculating plane. We shall show thai
tliis characteristic is precisely the tangent at the
point of r. It will be sufBcient to establish the eqnatioos
Adx + Bdy -^Cdz = 0, dA dx -^ dBdy + dC dM » 0.
The first of these is the first of (6), while the seoood is sasily
obtaineil by differentiating the first and then tw^inj qj^ q| ||^
second of (6). It follows that the characteristio is parallel to
the tangent, and it is evident that each of them peisos thioofh
the point (x, y, z)\ hence they coincide.
This method of forming the developable gives a elear idea of
the appearance of the surface. Let .4 B be an arc of a skew curve.
At each point M of AB draw the tangent, and cooaider only that
half of the tangent which extends in a certain direolkNi, — frooi A
toward B, for example. These half rays form one nappe 8i of the
developable, bounded on three sides by the arc AB and the taA>
gents A and B and extending to infinity. The other ends of the taa-
gents form another nappe .S', similar to S^ and joined to S, aloof Ibo
arc A B. To an observer placed above them thrae two nappes nppeor
to cover each other partially. It is evident that any plane not tan-
gent to r through any point O oi AB cuts the two nappes ^*i and 5,
of the developable in two branches of a curve which has a cusp at O,
The skew curve T is often called the tdge ^ r t f rt §t Um of the
developable surface.*
It is easy to verify directly the statement just made. Let us
take O as origin, the secant plane as the ary plane, the tangent to T as
the axis of z, and the osculating plane as the xs plana. Aanmiaf
that the coordinates x and y of a point of V can be expanded b poweie
of the independent variable «, the equations of T are of the form
a: = a,«* + at«*+ yo *.«•+•••,
for the equations
dM dM d^
• The English term " edge of l egiiM l oi i " does MS ■>!■■* •^ ^ T'^^^
of cusps. The French terme^srite del
are more suggestive. — Trajh.
464 SKEW CURVES [x; §222
must be satisfied at the origin. Hence the equations of a tangent
at a point near the origin are
2arz + '" ^b^z^■\-"'
Setting Z = 0, the coordinates X and Y of the point where the tan-
gent meets the secant plane are found to have developments which
begin with terms in z^ and in «*, respectively ; hence there is surely
a cusp at the origin.
Example. Let us select as the edge of regression the skew cubic « = <, y = <« ,
z = t*. The equation of the osculating plane to the curve is
(27) t«-3t2X+3<r-Z = 0;
hence we shall obtain the equation of the corresponding developable by writing
down the condition that (27) should have a double root in t, which amounts to
eliminating t between the equations
t2_2<X + r=0,
Xt^-2tY + Z = 0.
(28) I
The result of this elimination is the equation
(XF - z)2 - 4(X2 - r)(r2 - xz) = o,
which shows that the developable is of the fourth order.
It should be noticed that the equations (28) represent the tangent to the given
cubic.
222. Differential equation of developable surfaces. If « = F(a;, y) be
the equation of a developable surface, the function F(x, y) satisfies
the equation s^ — rt = 0, where r, s, and t represent, as usual, the
three second partial derivatives of the function F(x, y).
For the tangent planes to the given surface,
Z =:pX + qY + z-px-qyy
muet form a one-parameter family ; hence only one of the three
coefficients /?, q^ and z—px — qy can vary arbitrarily. In particular
there must be a relation between p and q of the form f{p, q) = 0.
It follows that the Jacobian D{py q)/D(x, y) = rt — s^ must vanish
identically.
Conversely, if F(x, y) satisfies the equation rt — s^ = 0, p and q
are connected by at least one relation. If there were two distinct
relations, p and q would be constants, F(x, y) would be of the form
ax-^by -^-Cy and the surface z = F(ic, y) would be a plane. If there
XI, 1228] ENVELOPES OF MiitACES 466
is a single relation between p and y, it may be writtm in the form
7 —f(p)f ^^^re p do69 not reduce to a eonstani. But we also hm9%
hence z — px — qy is also a function of p, lay f(p), wb«Mf«r
rt — 8^ z=0. Then the unknown function F(x, y) and it« ptH ial
derivatives p and y satisfy the two equations
y = <i>(P)f « -;w - ^rty - f(p).
Differentiating the second of these equations with retpect to x aod
with respect to y, we find
[^ + y <I>'(P) + 'A'C/')] ^ = 0, [x + y ^'Cp) -»- ^'(rt] ^ - 0.
Since p does not reduce to a constant, we must baro
hence the equation of the surface is to be found by alinuDittiig •
between this equation and the equation
«=iw; + y^p)-f f(/)),
which is exactly the process for finding Uie envelope of the lanjily
of planes represented by the latter equation, /> being thought of as
the variable parameter.
223. Envelope of a family of skew aurea. A one-paramcUr fiMily
of skew curves has, in general, no envelope. Let us onniidf finl
a family of straight lines
(29) X = «»+/», y = *«+ •
where a, b^ p^ and q are given functionii of a variable piiimtw «.
We shall proceed to find the conditions undor whifih orstj m a«bs t
of this family is tangent to the same skew conre T. Let « « 4(a)
be the z coordinate of the point ^1/ at which the rariable stvmifht
line D touches its envelope T. Then the roqaired curve T will be
represented by the equations (29) together with the eqnaiioo
z = <^(a), and the dire<^tion cosines of the tangvot lo T will be pto*
portional to dx/da, dyjda^ d*/da, i.e. to the three qotntitiee
a <^'(a) -f a'«^(a) -k- p\ ft^V) + *'♦(«> + f*. 4'(a\
466
SKEW CURVES
[XI, § 223
where a', b', p\ and q' are the derivatives of a, h, p^ and q, respec-
tively. The necessary and sufficient condition that this tangent be
the straight line D itself is that we should have
dx _ dz
da da
dy _ .d&
da da
that is,
a'4>{a) +1?' = 0, h'^{a) + g'' = 0.
The unknown function 4>{a) must satisfy these two equations;
hence the family of straight lines has no envelope unless the two
are compatible, that is, unless
a'q^ - Vp' = 0.
If this condition is satisfied, we shall obtain the envelope by setting
<l>(a)=-p'/a'=-q'/b'.
It is easy to generalize the preceding argument. Let us consider a
one-parameter family of skew curves (C) represented by the equations
(30)
F(x, y, «, a) = , ^(x, 2/, «, a) = 0,
where a is the variable parameter. If each of these curves C is
tangent to the same curve r, the coordinates (x, y, z) of the point
M at which the envelope touches the curve C which corresponds to
the parameter value a are functions of a which satisfy (30) and
which also satisfy another relation distinct from those two. Let
dx^ dy^ dz be the differentials with respect to a displacement of M
along C ; since a is constant along C, these differentials must satisfy
the two equations
(31)
(dF^ dF . dF. _
-^ dx -\- T- dy -\- -z- dz = 0,
ox Cy '' cz
-^ dx -{- 1^ dy -{- ^r- dz = .
^ox ^y ^«
On the other hand, let hx^ hy, Bz, Sa be the differentials of x, y, «,
and a with respect to a displacement of M along r. These differen-
tials satisfy the equations
(32)
' dF dF dF dF
d^
dz
ye.^^'P^^'i^^Tj-'
XI, §223] £NVKLOP£S OF SURFACES 467
The necessary and sufficient conditions that the earrm C tad r
be tangent are
dx _dy dm
ix " iy " $g*
or, making use of (31) and (32),
It follows that the coordinaiet (x, y, s) of the point o/temgmt^ Miuf
satisfy the equations
(33) F=0, ♦ = 0, |f = 0. g-0.
Hence, if the family (30) is to have an envelope, the four equattoos
(33) must be compatible for all values of a. Conversely, if tbcM
four equations have a common solution in a*, y, and c for all Taloee
of a, the argument shows that the curve T described by the point
(ar, y, z) is tangent at each point (x, y, x) upon it to the
ing curve C. This is all under the supposition that the ratios
dx^ dy, and dz are determined by the equations (31), that it, that tha
point (a;, y, z) is not a singular point of the curve C.
Note. If the curves C are the characteristics of a ODe-paramsIrr
family of surfaces F(x, y, ;:;, a) = 0, the equations (33) reduoe to
the three distinct equations
dF ^F
(34) F=0, —=-0, V^ = 0:
hence the curve represented by these equat ♦' . .• •' :■••
of the characteristics. This is the generali/u • ■ : i • i: - ' m
proved above for the generators of a developable sorteee.
The equations of a one-parameter family of straight
in the form
(36) _,___-,__.
where Xo, Vq> ^o^ ^^ ^, <" arc itincuonB of a variable paramelar a. U ls<
find directly the condition that thb family iboold have an
denote the common value of each of the preoedlof imtios ;
of any point of the straight line are given by the eqaslloiis
and the question is to determine whether it Is posrfhit to Mhrtllals lor I nea a
function of a that the variable straight line ahoald alwara remain tancaal lo
468
SKEW CURVES
[XI, § 224
the curve described by the pomt (x, y, z). The necessary condition for this is
that we should have
xii-^a'l ^ yh-^l/l ^ zf> + Cl
a b c
(36)
Denoting by m the common value of these ratios and eliminating I and m from
the three linear equations obtained, we find the equation of condition
(«n
x6
yo ^
a
h c
or
h' c'
= 0.
If this condition Lb satisfied, the equations (36) determine {, and hence also the
equation of the envelope.
III. CURVATURE AND TORSION OF SKEW CURVES
224. Spherical indicatrix. Let us adopt upon a given skew curve F
a definite sense of motion, and let s be the length of the arc AM
measured from some fixed point A as origin to any point M, affixing
the sign + or the sign — according as the direction from A toward
M is the direction adopted or the opposite direction. Let MT be
the positive direction of the tangent at M, that is, that which cor-
responds to increasing values of the arc. If through any point in
space lines be drawn parallel to these half rays, a cone S is formed
which is called the directing cone of the developable surface formed
by the tangents to F. Let us draw a sphere of unit radius about O
as center, and let 2 be the line of intersection of this sphere with
the directing cone. The curve % is called the spherical indicatrix
FiQ. 48
of the curve F. The correspondence between the points of these two
curves is one-to-one ; to a point Af of F corresponds the point m where
the parallel to A/r pierces the sphere. As the point Af describes the
Xl.$22fl] CURVATURE TORSION 4$%
curve r in the positive sense, the point m deecrtbee the eurre 1 ui
a certain sense, which we shall adopt as poeitiTt. Then IIm ooiit
sponding arcs « and <r inorease simultaaeously (Rig. 48).
It is evident that if the point O be displaoed, the whoU eorre S
undergoes the same translation ; hence we may suppoee tlMl O lies
at the origin of coordinates. Likewise, if the positive mom od tke
curve r be reversed, the curve 2 is replaced bj a curve syiBaMtrkal
to it with respect to the point O \ but it should be notioed thai ibe
positive sense of the tangent fiU to S is independent of tba mbm of
motion on T.
The tangent plane to the directing cone along the geoeialor Om it
parallel to the osculating plans at M, For, let AX -^ BY -i- CZ b
be the equation of the plane Omm'f the center O of the spheM
at the origin. This plane is parallel to the two tangents at if
at M' ; hence, if ^ and ^ + A are the parameter valuM which
spond to M and M\ respectively, we most have
(38) Af(t) + J5^'(0 -f Cf (0 = 0.
(39) Af'(t 4- A) + Bi^Xt + A) + C^'(t + A) - 0.
The second of these equations may be replaced by the equafcico
, f(t + h) -fit) ^ ^ »'« + h)- »'(o , ^ »'(> -f A) - »'(» , a^
AAA
which becomes, in the limit as A approaohM Mro,
(40) Af\t) + B^'\t) + C^'XO - 0.
The equations (38) and (40), which determine i4, B, and C for the
tangent plane at m, are exactly the same as the equattonc (6) which
determine A, 5, and C for the osculating plane.
225. Radius of curvature. Let • be the angle beiWMO the positive
directions of the tangenU MT and itV at two neighboring poialc
M and 3/' of T, Then the limit of the ratio •/arc J/JT, m M
approaches ilf, is called the eurvahtre of T at the point If. just as
for a plane curve. The reciprocal of the curvature is cdM Ika
radius of curvature : it is the limit of arc J/JT/ti.
Again, the radius of curvature R may be defined to be the limit
of the ratio of the two infinitesimal arM MMt and mm\ for we hava
arc AfAf' ^ arc JfiT' ^ aictam* ^ thotdwtm'
M arc mm' chord
470 SKEW CURVES [XI, §225
and each of the fractions (arc mm ')/(chord) mm^ and (chord mm')/i3i
approaches the limit unity as m approaches m'. The arcs5(=J/M')
and a-(=mm') increase or decrease simultaneously; hence
ds
(«) ^ = 1-.
Let the equations of r be given in the fotm
(42) x==f(t), y = <t>(t), z = ^{t),
where O is the origin of coordinates. Then the coordinates of the
point m are nothing else than the direction cosines of MT, namely
_ ^^ Q _ ^y — ^^
ds ds ^ ds
Differentiating these equations, we find
dsd^x —dxd^s .^ dsd^y — dyd^s , dsd^z — dzd^s
""= — dT^ — ' '^^^ ds' ' "y^ — s^
d.' = d<^ + d^ + dr' = S{ds<^x-dxdH)\
where O indicates as usual the sum of the three similar terms
obtained by replacing x by a;, y, z successively. Finally, expanding
and making use of the expressions for ds^ and ds d^Sj we find
_ Sdx^ S{d^xf-\Sdxd^xT
ds*
By Lagrange's identity (§ 131) this equation may be written in
the form
dxd^Zj
a notation which we shall use consistently in what follows. Then
the formula (41) for the radius of curvature becomes
(44) R*= "^'^
da* = y
ds*
where
(43)
r
= dyd^z-dzd^y, B = dz d^x
C = dxd^y — dyd^Xj
and it is evident that R* is a rational function of x, y, z, x\ y\ z\
«", y", «". Tlie expression for the radius of curvature itself is
irrational, but it is essentially a positive quantity.
XI, §226] CURVATURE TORhlUN 471
Note. If the independent Tariable aeleeted ia the tra « of Iho
curve r, the functions /(«), ^(«), and f («) ntiify the eqaatioo
Then we shall have
(46) " t/a =/"(«) (if, <//3 = ^"Wrfi, dy . f (•) A.
and the expression for the radius of ounrature iMumet tha pMtae-
ularly elegant form
(44') ;^. = [/"(')]• + [♦"(')]* + [*"(»)?•
Principal normal Center of curratnre. Let us draw a line
through 3/ (on F) parallel to mt, the tangent to 1 at m. Let MN
be the direction on this line which corresponds to the positlTe diree>
tion vit. The new line MN is called the principal normal to T at i# :
it is that normal which lies in the osculating plane, sinee wU is
perpendicular to Om and OnU is parallel to the oecolatinf piMM
(§ 224). The direction MN is called the pomUm dirmiiam ^ Os
principal normal. This direction is uniquely defined, since the posi-
tive direction of mt does not depend upon the choice of the positive
direction upon F. We shall see in a moment how the direetta hi
question might be defined without using the indicatiiju
If a length MC equal to the radius of corvatore at 1# be laid off
on iMX from the point 3/, the extremity C is called the
curvature of F at A/, and the circle drawn around C in the
ing plane with a radius MC is called the rireis of emrvoHtr*.
a\ p, y be the direction cosines of the principal nonnaL
coordinates (xi, yi, «i) of the center of curvature are
arj = X -f Ra\ yi = y + Rff* »i - » + Ry*'
But we also have
, da dads ^da , dsd^m - datd^e
da d* da d* d^
and similar formulae for ff and y'. Replacing a' by iU value in
the expression for x, we find
.dMd^x ^djtd^s
472 SKEW CURVES [XI, §226
But the coefficient of R^ may be written in the form
ds* "" ds*
or, in terms of the quantities A, B, and C,
Bdz — Cdy
ds^
The values of yi and z^ may be written down by cyclic permutation
from this value of Xi , and the coordinates of the center of curvature
may be written in the form
'' , ^^ Bdz — Cdy
x^ = x + R'' —. ^j
(46)
ds""
h C dx — A dz
2" = 2' + « 1? '
These expressions for x^ yi, and z^ are rational in x, y, z, x\ y\ z\
x", 2/", «".
A plane Q through M perpendicular to MN passes through the
tangent MT and does not cross the curve r at M. We shall proceed
to show that the center of curvature and the points of V near M lie
on the same side of Q. To show this, let us take as the independent
variable the arc s of the curve r counted from M as origin. Then
the coordinates X, Y, Z oi a point M ' of F near M are of the form
x^-
s dx s^ /d^x \
1 ^ "^ 172 V"^ 7
the expansions for Y and Z being similar to the expansion for X
But since s is the independent variable, we shall have
dx
d^x da da da- 1 ,
57="'
ds^ ~ ds ~ da- ds ~ R
and the formula for .Y l)ecomes
^=^ + - + (l'+')A
If in the equation of the plane Q,
aXX - ar) 4- p'( V - y) + y\Z - ;^) = 0,
XI, §227] CUBVATLKE T0R810N 47t
Xf Y, and Z be replaced by these expaoaiona in the left-hand wanibM
the value of that member ia found to be
where 17 approaches zero with «. Thia quantity ia poeittTe for all
values of s near zero. Likewiae, replacing (Jt, K, Z) by the eoSidi-
nates (x + Jia'j y + Rp^ z + Ry^ of the center of canntui% the
result of the substitution is R, which ia eaaentially poeitive. Heaee
the theorem is proved.
227. Polar line. Polar surface. The perpendicular A to the oeeu-
lating plane at the center of curvature ia called the polar iims. Thia
straight line is the characteristic of the normal plane to P. For, in
the first place, it is evident tliat the line of iiitnuectiou D oi the
normal planes at two neighboring pointa M and M* ia perpeodienlar
to each of the lines MT and M'T' ; hence it is also perpendicular to
the plane ynOm'. As M' approaches M, the plane mOm* approaches
parallelism to the osculating plane ; hence the line D approtehes a
line perpendicular to the osculating plane. On the other hand, to
show that it passes through the center of curratorey let « be the
independent variable ; then the equation of the normal plane is
(47) a(X - x) + ^(r- y) 4- y(Z - a) = 0,
and the characteristic is defined by (47) together with the equation
(48) ^(.Y -x)-f|V-y) + ^'(^ -«)-!- 0-
This new equation represents a plane perpendicular to the principal
normal through the center of curvature; hence the intsneetkn of
the two planes is the polar line.
The polar lines form a ruled surface, which ia called the ^slnr
surface. It is evident that this surface ia a developable, atnee we
have just seen that it is the envelope of the normal plane to T.
If r is a plane curve, the polar surface ia a oylinder whoea right
section is the evolute of T ; in this special case the preceding
ments are self-evident.
228. Torsion. If the worda "tangent line" in the daOBitkn of
curvature (§ 225) be replaced by the words «oeeulatinf ptsB^" »
new geometrical concept ia introduced which mmtnim, in a
the rate at which the osculating plane tuma. IM «' be the angle
between the osculating planea at two neighboring poinU J/ and if';
474 SKEW CURVES [XI, §228
then the limit of the ratio <o'/arc MM', as M approaches M', is called
the torsion of the curve r at the point M. The reciprocal of the
torsion is called the radius of torsion.
The perpendicular to the osculating plane at M is called the
binormal. Let us choose a certain direction on it as positive, — we
shall determine later which we shall take, — and let a", j8", y" be
the corresponding direction cosines. The parallel line through the
origin pierces the unit sphere at a point n, which we shall now put
into correspondence with the point M of r. The locus of w is a
spherical curve 0, and it is easy to show, as above, that the radius
of torsion T may be defined as the limit of the ratio of the two corre-
sponding arcs MM' and nn' of the two curves F and ®. Hence we
shall have
d
dr"
where t denotes the arc of the curve ©.
The coordinates of n are a", )8", y", which are given by the formulae
(§ 215)
a"=— ^=4_, p"= , ^ =:> y' = , ^
±Va^+¥+c^ ±\/a^+b^-\-c^ ±V^2+i52-fc«
where the radical is to be taken with the same sign in all three
formulae. From these formulae it is easy to deduce the values of
da"y dp'\ rfy"; for example,
^^„ ^ . (A' -\-B^ + C^dA-A{A dA -hBdB-^CdC)
(A^ + B^-\- C^)^
whence, since dr^ = da"^ + dp"^ -f dy"%
,^.^ SA^SdA^-[S(AdA)r ^
(A^ + B^ -{- Cy
or, by Lagrange's identity,
SiBdC-CdBf
{A^ + B^-^ Cy •
where o denotes the sum of the three terms obtained by cyclic per-
mutation of the three letters A, B,C. The numerator of this expres-
sion may be simplified by means of the relations
Adx+ Bdy+ Cdz = 0,
dA dx -f dBdi/ + dCdz== 0,
whence
1
^'"^ BdC-CdB CdA^AdC AdB-BdA
-F
^^ 5"«j CLHVATURE TORSION 475
where /C is a quantity deBned by the equatioii (49) itatll This gitct
where /T is defined by (49) ; or, expanding,
= S{dzd*xd*ij-~dxcPMd*y),
where aS* denotes the sum of the three terms obtained by oyelie per-
mutation of the three letters x, y, z. But this value of IT it cxaelly
the development of the determinant A [(8), S 216]; hence
and therefore the radius of torsion is given by the formula
(50) r=±ill±^l±^.
A
If we agree to consider T essentially positive, ac we did the radios
of curvature, its value will be the absolute value of the teoocid OMOi*
ber. But it should be noticed that the expression f or T is ratk»al
in X, y, «, x'j y\ «', x", y", «"; hence it is natural to represent the
radius of torsion by a length affected by a sign. The two sifM
which T may have correspond to entirely different aspects of Um
curve r at the point M.
Since the sign of T depends only on that of A, we shall iiiTestigstt
the difference in the appearance of T near M when A has different
signs. Let us suppose that the trihedron Oryt is placed so thai an
observer standing on the xy plane with his fsel at O and his hsnd in
the positive z axis would see the x axis turn throogh W t§ kkl^
if the X axis turned round into the y axis (see fooCnole, p. 477).
Suppose that the positive direction of the binormal MS^ has been so
chosen that the trihedron fonned from the lines If T, MN, MS^ has
the same aspect as the trihedron formed from the lines Om^Op,OB;
that is, if the curve T be moved into such a position thai JT eoineides
with O, i\fT with Ox, and MN with Oy, the dirsetioo MN^ will eote*
cide with the positive z axis. During this motion the absolvis Talne
of T remains unchanged ; hence A cannot Tsnish, and hioee it
476
SKEW CURVES
[XI, § 228
even change sign.* In this position of the curve r with respect to
the axes now in the figure the coordinates of a point near the origin
will be given by the formulae
(61)
B
where c, c', e" approach zero with f, provided that the parameter t is
so chosen that ^ = at the origin. For with the system of axes
employed we must have dy = dz = d^z = when ^ = 0. Moreover
we may suppose that a^ > 0, for a change in the parameter from t to
— t will change ai to — aj . The coefficient b^ is positive since y must
be positive near the origin, but c^ may be either positive or negative.
On the other hand, f or ^ = 0, A = 12a^h<^c^ dt^. Hence the sign of A
is the sign of c^. There are then two cases to be distinguished. If
c, > 0, X and z are both negative f or — A < ^ < 0, and both positive
for < < < A, where A is a sufficiently small positive number ; i.e.
an observer standing on the xy plane with his feet at a point P on
:n
N
'M'
Fia. 49, a
M
Fig. 49, 6
\M'
the positive half of the principal normal would see the arc MM^ at
his left and above the osculating plane, and the arc MM" at his right
below that plane (Fig. 49, a). In this case the curve is said to be
sinistrorsal. On the other hand, if Cg < 0, the aspect of the curve
would be exactly reversed (Fig. 49, b), and the curve would be said
to be dextrorsal. These two aspects are essentially distinct. For
example, if two spirals (helices) of the same pitch be drawn on the
same right circular cylinder, or on two congruent cylinders, they
will be superposable if they are both sinistrorsal or both dextrorsal ;
but if one of them is sinistrorsal and the other dextrorsal, one of
them will be superposable upon the helix symmetrical to the other
one with respect to a plane of symmetry.
* It would be easy to show directly that A does not change sign when we pass from
one set of rectangular axes to another set which have the same aspect.
XI, $229] CURVATURE TORSION 477
In consequence of these rest
(52) r = -
In consequence of theee retolto we ihall wrile
A
i.e. at a point where the curve is dextrorsal T ihall be poeitive, while
T shall be negative at a point where the curve is sinistrofiiL A dif>
ferent arrangement of the original ooOrdinate trihedron OmgM would
lead to exactly opposite results.*
229. Frenet'8 formulfe. Each point if of T is the vertex of a tii-
rectangular trihedron whose aspect is the same as that of the trib^
dron Oxi/z, and whose edges are the tangent, the principal Doraal,
and the binormal. The positive direction of the principal normal is
already fixed. That of the tangent may be chosen at pleasure, but
this choice then fixes the positive direction on the binonnaL The dif*
ferentials of the nine direction cosines (a, /J, y), (a\ /J*, y*), (a", /f , y*^
of these edges may be expressed very simply in terms of R, T, and
the direction cosines themselves, by means of certain fonnuls doe
to Frenet-t We have already found the formuls for da^ dfi, and dy:
^^"^^ ds R ds R ds R
The direction cosines of the positive binormal (§ 228) are
a"=e-;=4==, fl" = «-==J=, y*'«€ , ^ i>
^A^-hB^+c^ y/A*T^Tc* Va*V¥Tc^
where c = ± 1. Since the trihedron (MT, MN, MN^) has the «MM
aspect as the trihedron Oxy«, we must have
.' = rr-^r", or <'-'* ^XZc '
On the other hand, the formula for da" may be writteo
„^ BjBdA -Adm-^CjCdA -AdC)
(A*-^B»+C^^
or, by (49) and the relation /T = A,
da" ^ C/8~By ^ g^A
• It is usual in America to adopt aa anaagWBMt «f
described above. Hence we •bookl write r« ->• U' + ^ + C^/A, «a.
the footnote to formula (54), §2».— TaAJia.
t NouvelUs Annate* de IfalAtfrnoNfUM, l»«. ^ »i>
478 SKEW CURVES [XI, §229
The coefficient of a' is precisely 1/T, by (52). The formulae for
rfj3" and dy" may be calculated in like manner, and we should find
(^) IT^J' -dT-f' ds -r'
which are exactly analogous to (53).*
In order to find da\ dp'j dy\ let us differentiate the well-known
formulae
aa' + )8)3' + yy' =0,
a'a" + ^')8"+y'y" = 0,
replacing da, dp, rfy, rfa", dff\ c^y" by their values from (53) and
(54). This gives
a'da' + p'dp' + y'dy =0,
ds
a da'+/3 dp + y ^^y' + — = 0,
d<i
a''da' + P"dp-\-y"dy-hj; = 0',
whence, solving for da', dp, dy\
da'
a a"
dp p p"
dy'_
_z_y:.
ds
R t'
ds R t'
ds
R T
(65)
The formulae (53), (54), and (55) constitute Frenet's formulae.
Note. The formulae (54) show that the tangent to the spherical
curve described by the point n whose coordinates are a", p\ y" is
parallel to the principal normal. This can be verified geometrically.
Let S' be the cone whose vertex is at and whose directrix is the
curve 0. The generator On is perpendicular to the plane which is
tangent to the cone S along Om (§ 228). Hence S' is the polar cone
to S. But this property is a reciprocal one, i.e. the generator Om
of S is surely perpendicular to the plane which is tangent to S'
along On. Hence the tangent mt to the curve 2, since it is perpen-
dicular to each of the lines On and Om, is perpendicular to the
plane mOn. For the same reason the tangent nt' to the curve © is
perpendicular to the plane mOn. It follows that mt and nt' are
parallel.
• If we had written the formula for the torsion In the form 1/7'= A/(^« + B* + C^,
Frenet's formulw would liave to be written in the form da^'/da -- a'/T, etc
[Hence this would be the form if the axes are taken as usual in America. —Trans.]
XI, $230]
CURVATURE TORSIOH
479
230. Expansion of z, y, and s la powtrt of a. Oiren two fttselioiit
R = <^(.«;, T = ^(«) of an independunt variable «, the fim of whiflh
is positive, there exists a skew curve r whieh is oompUUlj ^••ntii
except for its position in space, and whooe radius of curvature and
radius of torsion are expressed by the given equations in tetMs of
the arc s of the curve counted from some fixed point upon it A rig*
orous proof of this theorem cannot be given until we have riisnisoed
the theory of differential equations. Just now we shall maroly show
how to find the expansions for the coordinates of a poiol on tlM
required curve in powers of «, assuming that such esptatioos tziii.
Let us take as axes the tangent, the principal aomial, and tho
binormal at 0, the origin of arcs on T. Then we shall have
/~1 \d8 J 0^ 1.2 \dsV» 1.2. S\ds»/»"'
where Xy y, and z are the coordinates of a point on T, But
dx
ds
= a
d^x
ds*
da
ds
a
r'
whence, differentiating,
In general, the repeated application of Frenet's formula
where i,, Af,, P, are known functions of R, T, and their soooeetiTe
derivatives with respect to #. In a similar maniior the aurositivo
derivatives of y and « are to be found hf replacing («, a', O ^
(A fiy P") and (y, y', y"), respectively. But we have, at the origiBt
«o = 1, ^0 = 0, yo = 0, ai = 0, /aj«= 1, yi-O. of -0, flf-O, >C-li
hence the formulae (66) become
I- .
^ = 1" 6R' ^ '
t* i^ dR
(66') {y^ 2R'eR*ds
«»
6RT
480 SKEW CURVES [xi,§23i
where the terms not written down are of degree higher than three.
It is understood, of course, that R, T, dR/dSj • • • are to be replaced,
by their values for s = 0.
These formulae enable us to calculate the principal parts of cer-
tain infinitesimals. For instance, the distance from a point of the
curve to the osculating plane is an infinitesimal of the third order,
and its principal part is — s^/6R T. The distance from a point on
the curve to the x axis, i.e. to the tangent, is of the second order,
and its principal part is s^/2R (compare § 214). Again, let us cal-
culate the length of an infinitesimal chord c. We find
c'' = x^ + 2/» + s^ = »'-j— + •••,
where the terms not written down are of degree higher than four.
This equation may be written in the form
which shows that the difference s — c is an infinitesimal of the
third order and that its principal part is s'/24i^l
In an exactly similar manner it may be shown that the shortest
distance between the tangent at the origin and the tangent at a
neighboring point is an infinitesimal of the third order whose prin-
cipal part is s^/12RT. This theorem is due to Bouquet.
231. Involutes and evolutes. A curve Fj is called an involute of a
second curve F if all the tangents to F are among the normals to Fj ,
and conversely, the curve F is called an evolute of Fj . It is evident
that all the involutes of a given curve F lie on the developable sur-
face of which F is the edge of regression, and cut the generators of
the developable orthogonally.
Let (x, 7/, z) be the coordinates of a point M of F, (a, yS, y) the
direction cosines of the tangent MT, and I the segment MM^ between
M and the point Afi where a certain involute cuts MT. Then the
coordinates of Mi are
xi = xi-lay yi'^y-hip, «i = « + hf
whence
dxi = dx -\-lda -\- adl,
dyx = dy'hldp + pdl,
dZi = dz + Idy -\- ydl.
XI, §231] CrRVVTiMV 'OKSlOir 491
In order that the curve deM;ribd(l by i/| thould be nflnftnl to 111/
it is necessary and sufficient that a dx^ -^ fidy^ 4- T^i «I hm iJi! vaftialL
i.e. that we should have
adx-\- fidy -^ydM + dl-^Hada-^ fidfi + ydy)mO,
which reduces to ds -^tUsaO, It foUowi thai the mvoliilii to a
given skew curve F may be drawn by the tame oonftnMlMNi whUk
was used for plane curves (§ 206).
Let us try to find all the evolutes of a
given curve F, that is, let us try to pick
out a one-parameter family of uormab to
the given curve according to some contin-
uous law which will group these normals
into a developable surface (Fig. 50). Let
Z> be an evolute, <f> the angle between the
normal MMi and the principal nonual MN,
and I the segment MP between Ai and the
projection P of the point Mi on the principal nonnaL TImb tl»
coordinates (xi, y^ z{) of 3f| are
X, = ar4-/a' + /<T"tan^,
(67)
{X, = ar + '<t' + la" tan ^,
yi = y + //9' + //r'tan4,
«j=« + /y' + /y"tan^,
as we see by projecting the broken line MPMi upoii the thrM aiM
successively. The tangent to the canre described by the point y%
must be the line MM^ itself, that is, we must hare
dxx ^^i ^
X, - X ■" y, - y *i - «
Let k denote the common value of these ratioe ; then the e o n d Hi ea
dxi = ik(xi - x) may be transformed, by inserting the Tiluee ol j^
and dxi and applying Frenet's formulie, into the form
arf*(l - ;^) + a'(rf/ + /tan 4^ - «)
+ a"[rf(/tan^) - ^ - A/tan^] - 0.
The conditions rfyi =: A: (y» - y) and rf^h - *(«t - ») W^ to earthy
similar forms, which may be deduced from the preoediag by n^^Mr
cing (a, a', a") by ()9, ff. /T) and (y. y', yO» wn****^/- »*»« *^
482 SKEW CURVES [XI. §231
determinant of the nine direction cosines is equal to unity, these
three equations are equivalent to the set
(58)
ds
dl -\-l tan <ft — = kl,
Ids
d(l tan <^) — - = kl tan <f>.
From the first of these I — R, which shows that the point P is the
center of curvature and that the line PM is the polar line. It fol-
lows that all the evolutes of a given skew curve T lie on the polar sur-
face. In order to determine these evolutes completely it only remains
to eliminate k between the last two of equations (58). Doing so
and replacing Ihy R throughout, we find ds = T d<f}. Hence <^ may
be found by a single quadrature :
(59) * = *« +X f '
If we consider two different determinations of the angle <^ which
correspond to two different values of the constant <^o> the difference
between these two determinations of <^ remains constant all along r.
It follows that two normals to the curve T which are tangent to two
different evolutes intersect at a constant angle. Hence, if we know
a single family of normals to r which form a developable surface,
all other families of normals which form developable surfaces may
be found by turning each member of the given family of normals
through the same angl^, which is otherwise arbitrary, around its
point of intersection with r.
Note I. If r is a plane curve, T is infinite, and the preceding
formula gives <f> = <f>Q. The evolute which corresponds to </)o = is
the plane evolute studied in § 206, which is the locus of the centers
of curvature of r. There are an infinite number of other evolutes,
which lie on the cylinder whose right section is the ordinary evo-
lute. We shall study these curves, which are called helices, in the
next section. This is the only case in which the locus of the cen-
ters of curvature is an evolute. In order that (59) should be satis-
fied by taking </> = 0, it is necessary that 7' should be infinite or
that A should vanish identically ; hence the curve is in any case a
plane curve (§216).
XI, §232] CURVATURE T0B8I0H Ut
Note II. If the ourre Z> is an e volute of r, it follows UmI F it ab
involute of D. Honoe
</«i-rf(jirAr,),
where «i denotes the length of the arc of the orolute eomltd f rf
some iixed point. This shows that all the erolntM of utf fivw
curve are rectiiiable.
288. Helic«t. Let C be any plane curve and lei us la j off oa Um
ular to the plane of C erected at any point m on C a InfUi aiif
the length of the arc o- of C counted from aome find point A. l^ea iIm ihsv
curve r described by the point M ia called a kdiz. Lai « uIm ibt pteae «C C
aa the xy plane and let
be the coordinates of a point m of C In terns of the are v. Thm Iht ooftjl
nates of the corresponding point Jf of the onnre T wHl be
m »=/(<') . y = ^(<r). t = Jr<r,
where K is the given factor of proportionality. The f unciloas / and # «Urfy
the relation /'« -f 0'« = 1 ; hence, from (60),
dS« = (/'« + ^'« + jr«)da« = (1 + Jr«)d«r*.
where a denotes the length of the arc of r. It follow* that • a v VTTT* 4 If*
or, if s and o* be counted from the same point ^ on C, « s 4r Vi-F Jn,siaesH » t^
The direction cosines of the tangent to r are
(61) a=-^m., p^4^. -r.-T^^-
Since 7 is independent of cr, it is evident that the tangent to T
angle with the z axis ; this property ia charaoterkak : Jny
makes a constant angle toith afixtd ttraigkt Urn ia a Aelic In order to prvf*
this, let us take the z axis parallel to the given aumlgbt Una, and lai C be iha
projection of the given curve T on the cy plane. TIm eqvatloaa of T aai
be written in the form
(62) «=/(<r), |f = f(«r). m^H^^
where the functions / and ^ satisfy the relatloB /'• + #'» = 1, for tWs
amounts to taking the arc cr of C as the ludspaidsni varlabla. It
hence the necessary and sufBcient condition thai 7 ka eoMtiat Is ihaif'
be constant, that is, that ^(<r) should be of the ior« Xr + s». I* ' "
the equations of the curve V will be of the fona (•©) If iba
the point i = 0, y = 0, t = to-
Since 7 is consuni, the formuU dy/da « Y/M sbowt th atV _
principal normal is perpendicular to the fsiisnlnfs ef the QrttaAsr. Steea k Is
also perpendicular to the tangent to the balls, H la 1
therefore the osculating plane is nonnal to the ejl
484 SKEW CURVES [xi,§232
binormal lies in the tangent plane at right angles to the tangent to the helix ;
hence it also makes a constant angle with the z axis, i.e. 7" is constant.
Since 7' = 0, the formula d77d« = - 7/B - 7"/ T shows that 7/JB + Y' / T = 0;
hence the ratio T/B. is constant for the helix.
Each of the properties mentioned above is characteristic for the helix. Let
us show, for example, that eo&ry curve for which the ratio T/B is constant is a
heliz. (J. Bertrand.)
From Frenet's formulae we have
da _dp_ _ dy_ _ ^ _ J_ ,
da'' ~ d/3" ~ d7" ~ B~ H*
hence, if ^ is a constant, a single integration gives
a" = na-A, ^' = n^-B, 7" = H7 - C,
where A, By C are three new constants. Adding these three equations after
multiplying them by a, /S, 7, respectively, we find
Aa -\- Bp -h Cy = fl",
or
Aa -\- Bp -\- Cy H
V^2 + 52 4. (72 V^2 + 52 _|. era
But the three quantities
A B
are the direction cosines of a certain straight line A, and the preceding equa-
tion shows that the tangent makes a constant angle with this line. Hence the
given curve is a helix.
Again, let us find the radius of curvature. By (53) and (61) we have
whence, since 7' = 0,
This shows that the ratio (1 + K^/B is independent of K. But when ^ =
this ratio reduces to the reciprocal 1/r of the radius of curvature of the right
section C, which is easily verified (§ 205). Hence the preceding formula may
be written in the form B = r{l + K^)^ which shows that the ratio of the radius
of curvature of a helix to the radius of curvature of the corresponding curve C
is a constant.
It is now easy to find all the curves for which B and T are both constant.
For, since the ratio T/B is constant, all the curves must be helices, by Bertrand's
theorem. Moreover, since jB is a constant, the radius of curvature r of the
curve C also is a constant. Hence C is a circle, and the required curve is a
helix which lies on a circular cylinder. This proposition is due to Puiseux.*
• It is assumed in this proof that we are dealing only with real curves, for we
asBumed that A^ + li^ -^ C* does not vanish. (See the thesis by Lyon : Sur les
courbea a torsion constante, 1890.)
XI, f 233] CURVATURE TORSION
S88. B«rtnnd'a cottm. Hm prfaieipal
principal noriualii to an Inflaita munbtr ci
given curre. J. Beitnod tttooiptod to iad in
curyes whose principal nonnali are the prineipal ■ofik lo » 099m dbtm
ourre r. Let the ooOrdinatei x, y, s of a point of r be givea ae fttailaM el ifca
arc f. Let ua lay off on each prindpal nonnal a eegaaat of \m^h t, wmi hi tt*
coordinates of the extremity of this wginwn be X, F, 2 ; thm w iMI hmt
(64) X = x + ta', T^w-^lfiT, ZmM^kf,
The necessary and sufficient condition that the priDeipal nnr— I lollMcwf* r*
described by Uie point (JT, F, Z) ibould eolndde with the prtBdyid awal to r
is that the two equations
o'idYd^z - dZdi^T) + /r (dz<««x - dzd^z) + v(dx#r - tfr#X) • #
should be satisfied simultaneously. Tha nuanhif of aaoli of tiMM oqpMlioM li
evident. From the first, di = ; hence the lengtli of tiM eag— tTibo^M hi a
constant. Replacing dJT, (i*X, dY, • • • In the aeoood oqaatkm by tMr
from Frenet*s formulsB and from the fonnoUs obtataad by
Frenet's, and then simplifying, we finally find
M(-i)-(- !)'(»•
whence, integrating,
(66) i"*"?"*'
where V is the constant of intagratloo. It foUowa thai CJU rsgnirvrf c
those for which there entU a linear rtUiUon betm$m tk§ tmn tl hn md $k
On the other hand, it is easy to ahow that thia eoa dit i w i la adMnft
the length I is given by the relation (66).
A remarkable particular case had already baas ooHod by Italfi
that in which the radius of currature Is a conaianu In that eoaa (li|
I = R, and the curve r' defined by the eqoatlooa (64) la tba b»M of tba
of curvature of r. From (64), ■winrtng ImB
dx=^^a^^d., dr=^?,frdM. ^ — *y
which show that the tangent to r la Iha polar Una of T. Tba
ture ii' of r' is given by the formula
^ d x*.f<r»-i><p _-.
hence /T ah» la constant aad equal to R. Tba relauon ^ tha two
r and r' is therefore a raotpffooal ono: each of th«a la ••
the polar surface of the other. It la easy to mtfy iMh oi u>r^
the particular caaa of tha olroolar halix.
486 SKEW CURVES [XI, §234
Note. It is easy to find the general f ormulse for all skew curves whose radius of
curvature is constant. Let R be the given constant radius and let a, /3, y be any
three functions of a variable parameter which satisfy the relation a* + /S^ + 72 = l.
Then the equations
(66) X = U fader, T=Rfpd(r, Z = Rfyd(T,
where da = Vda^ + d^ + dy^, represent a curve which has the required prop-
erty, and it is easy to show that all curves which have that property may be
obtained in this manner. For a, /3, 7 are exactly the direction cosines of the
curve defined by (66), and a is the arc of its spherical indicatrix (§ 225).
IV. CONTACT BETWEEN SKEW CURVES
CONTACT BETWEEN CURVES AND SURFACES
234. Contact between two curves. The order of contact of two
skew curves is defined in the same way as for plane curves. Let T
and r' be two curves which are tangent at a point A. To each point
M of r near A let us assign a point M' of V according to such a law
that M and M' approach A simultaneously. We proceed to find
the maximum order of the infinitesimal MM' with respect to the
principal infinitesimal AM, the arc of T. If this maximum order
is n -\- 1, we shall say that the two curves have contact of order n.
Let us assume a system of trirectangular * axes in space, such
that the yz plane is not parallel to the common tangent at A, and
let the equations of the two curves be
(F) 1^ = ^(^)' (F') r = ^(^)^
^ ^ \z=<^(x), ^ ^ \z = ^(x).
If a^oj 2/0 ^0 are the coordinates of A, the coordinates of M and ilf '
are, respectively,
\xo + h, f(xo + h), <f>(xo + h)-] , [xo + k, F(xo + k), *(xo + A;)] ,
where A; is a function of h which is defined by" the law of corre-
spondence assumed between M and M' and which approaches zero
with h. We may select h as the principal infinitesimal instead of
the a,TO AM (§ 211) ; and a necessary condition that MM' should
be an infinitesimal of order n -f- 1 is that each of the differences
k-h, F(x,^k)-f(x,-\-h), ^xo-hk)-<l>(xo + h)
• It is easy to show, by passing to the formula for the distance between two points
in oblique coordinates, that this assumption is not essential.
XI, |2M] CnVTArT ^gj
should be an infinitatimal of uidia a 4- 1 or mor«. It foUovi Ikat
we must have
♦(«, + *)- ♦(jr. + A)- yA-
where a, /3, y remain finite as A approtfihet saia ^t''~^H ^ kv
its value h + aA"^* from the first of these eqnalioiis, the latter two
become
n^ + A + aA-^') -y)[^ + A). /8A-»,
♦(a^ + A + aA-*«) - ^a^ + A) - yA-».
Expanding F(io + A -f aA-"^*) and ♦(x, 4- A + aA-*«) by Tajlor^s
series, all the terms which contain a will have a factor A*^'; l^tuffiit
in order that the preceding condition be aatialled, mA oC tte
differences
^(^0 + A) -f{x^ + A), ♦(x, + A; - ^x^ + A)
should be of order n + 1 or more. It follows that if Jflf' is of
order ?i + l* the distance MN between the points Mi and N of IIm
two curves which have the same abscissa «^ + A will be at IsasC of
order n + 1. Hence the maximum order of tlie iBSaitesiaal ia
question will be obtained bj/ putting into c o rrwp on dintt tAs pmrnls
of the two curves which have the mtmo ab§eis9n.
This maximum order is easily evaluated. Since the two eurvsiB
aie tangent we shall have
/(xo) = F(Xo), /'(x^) = F'(xO, 4(«W)-*(^). 4Yx^^-#Y«J.
Let us suppose for generality that we also hare
fXxo) = F"(xo), , /^>(^) - P^(^h
but that at least one of the diflereneet
does not vanish. Then the distance MM* will be of ovder • «f 1
and the contact will be of order n. This result may also be siMlod
as follows : To find the ordsr ^ eontaet ^ Ups mw^tm V mmd P, mm-
sider the two teU of jtrofoetums (C, CT) mmd (C„ CQ ^f tks §imm
curves on the xy plane and the xa pUmo, ntpwe ii wo lp, mmd /W lAs
order of contact of each set; thm lAe mrdmr ef emtmet ^ tkm fhmm
curves T and T* vfiU be tho swuUUr ^ Mess Hso.
488 SKEW CURVES [XI, §235
If the two curves r and r' are given in the form
(r) x=f{t), y = <t>(t), z = ^(t),
(V) X=f{u), Y=^(u), Z = ^(u),
they will be tangent at a point u = t = tQii
H^o) = <i>(to) , *'(^o) = <f>'(to) , *(^o) = K*o) , ^'(M =^ 'A'(^o) .
If we suppose that f'(^o) '^s not zero, the tangent at the point of
contact is not parallel to the yz plane, and the points on the two
curves which have the same abscissa correspond to the same value
of t. In order that the contact should be of order n it is neces-
sary and sufficient that each of the infinitesimals ^(t) — <f>(t) and
"^(t) — ^(t) should be of order n -\-l with respect to t — t^, i.e. that
we should have
*'(^o) = «A'(^o), • • •, ^^^XM = «A^"H^o),
and that at least one of the differences
should not vanish.
It is easy to reduce to the preceding the case in which one of the
curves V is given by equations of the form
(67) x=f{t), y = <f>(t), z = ^{t),
and the other curve r' by two implicit equations
F{x,y,z) = 0, F^(x,y,z) = 0.
Eesuming the reasoning of § 212, we could show that a necessary
condition that the contact should be of order ti at a point of F
where t=:tQ is that we should have
,gg- (F(<o) = 0, F'(<„) = 0, ..., F«(<„) = 0,
^ -* ^F.(<„) = 0, F!(<o) = 0, •••, Fi">(<o) = 0,
where
m = nf(t}, *(o> "A(0] . F, (0 = F, if{t), ^(t), i,(t)-] .
235. Osculating curves. Let T be a curve whose equations are
given in the form (67), and let V be one of a family of curves in
2n -h 2 parameters a^by c, •', I, which is defined by the equations
(69) F{x, y,z,a,b,->-,l) = 0, F, (x, y, z, a, b.Cy^- -, I) = 0.
XI, §238] CONTACT
lu general it it possible to determino th« 2« 4- 2 [MfimHis bi mok
a way that the corresponding ourve V* has ooDUci ol order • vilb
the given curve F at a given point The ennre thus dfrtarf jned is
called the otculating curve of the familj (69) to the entire T. The
equations which determine Uie values of the ptxuietatt «, it i^ • • •, I
are precisely the 2m + 2 equations (68). It should be Doled
these equations cannot be solved unless eeoh of the fnnntinai F \
h\ contain at least n -f 1 parameters. For example, If the
F' are plane curves, one of the equations (69) oontams onlj three
parameters; hence a plane curve cannot have eootMi of
higher than two with a skew curve at a point takeo at
the curve.
Let us apply this theory to the simpler dasses of eanreSv — the
straight line and the circle. A straight line depends oo foor paia^-
eters ; hence the osculating straight line will have oootaei of the
first order. It is easy to show that it coincides with the
for if we write the equations of the straight line in the form
the equations (68) become
where (xo, yo, «o) is the supposed point of eootafli on P. Solriaf
these equations, we find
a = —
- <
*♦ ■• ^
which are precisely the values which give the taafent A
sary condition that the tangent should have eootael of the
order is that xH = a«i', y;' = &«;', that is,
ri yi H
The points where this happens are those disouased in I 21T.
The family of all circles in spaoe depends oa six paiMMian;
hence the asculaHng eireU will hare oootaei of the eeeood om^.
Let the equations of the circle be written in the form
F(«,y,«) = v4(«-a) + B(jf-*) + C(e-#) -0.
F»(«,y,*) = («-a)« + (y-*)« + («-«)"-^-<>.
490 SKEW CURVES [xi,§236
where the parameters are a, b, c, R, and the two ratios of the three
coefficients A, B, C. The equations which determine the osculating
circle are
A(x - a) + B(y - i) + C(z - c) = 0,
(X - ay + (y- by + (z- cy = R\
^ dx , , ,^dy , , ^dz
(x-a)^ + (S,-i)^ + (.-c)^ + f, = 0,
where x, y, and z are to be replaced by /(^), </>(^), and j/r(^), respec-
tively. The second and the third of these equations show that the
plane of the osculating circle is the osculating plane of the curve r.
If a, J, and c be thought of as the running coordinates, the last
two equations represent, respectively, the normal plane at the point
(ic, 2/, «) and the normal plane at a point whose distance from
(x, y, «) is infinitesimal. Hence the center of the osculating circle
is the point of intersection of the osculating plane and the polar
line. It follows that the osculating circle coincides with the circle
of curvature, as we might have foreseen by noticing that two curves
which have contact of the second order have the same circle of
curvature, since the values of y\ z\ y", «" are the same for the two
curves.
236. Contact between a curve and a surface. Let 5 be a surface
and r a curve tangent to 5 at a point A. To any point M of F
near A let us assign a point M' of S according to such a law that
M and M' approach A simultaneously. First let us try to find what
law of correspondence between M and 3/' will render the order
of the infinitesimal MM^ with respect to the arc AM 2u maximum.
Let us choose a system of rectangular coordinates in such a way
that the tangent to V shall not be parallel to the yz plane, and that
the tangent plane to .V shall not be parallel to the z axis. Let
(2^0) 2/0? ^c) ^6 the coordinates of ^1 ; Z — F{x, y) the equation of S-,
y =/(a;), z = <ti(x) the equations of T ; and w + 1 the order of the
infinitesimal MM' for the given law of correspondence. The
Xl,5'i3fij CONTACT 491
coordinates of 3/ are [a^ + A, /(as, -h A), ^g^ -♦-*)). Ltl X; Ft Mid
^ = F(A', r) be Uie oo6rdiiuitet of A/'. In otdet tbat i#jr<
be of order n -\- 1 with respect to the arc .1 Jl#, or, what
the same thing, with respect to A, it U ncirwiir/ that f4 t el Um
differences X — x^ Y — y^ and ^ — s should be an iiiftiiiieaiMal ai
least of order n + 1> that is, that we should have
where a, fiy y remain finite as A approaobaa xero. Heiioa wt shall
have
F(x 4- aA-^S y + /?*"''*) - « = yA''^*,
and the difference F(x, y) — « will be itself at least of order a -f 1.
This shows that the order of the infinitesimal MS, where H is tha
point where a parallel to the z axis pierces the 8iirfaoa» will bt al
least as great as that of MM'. The maximum order of <<^tti*t —
which we shall call the order of contaet of the curve and ike mtffmtm
— is therefore that of the distance MN with respect to the are AM
or with respect to h. Or, again, we may saj that the order of eoih
tact of the curve and the surface is the order of eomiaei htiwtem T
and the curre T' in which the surface S U cut hy ike ejfUmier wkitk
projects r upon the xy plane. (It is erident that the m axis lai^ be
any line not parallel to the tangent plane.) For the equatiooa of
the curve r' are
y=/(x), ;7=F[x,/l[x)] -♦(*),
and, by hypothesis,
♦(x„) = <^(ar«). ^'(r,) = ^'(r.).
If we also have
the curve and the surface have oontaet of order a, Siiiee the eqaa^
tion <P(x) = 4^(x) gives the abeoiassB of the poinU of iateieaetiea of
the curve and the surface, these conditions for eootaei of
at a point A may be expressed by saying that the onnre
surface in n -f- 1 coincident pointa at A.
Finally, if the curve T is given by equations of the form » -/^rV.
y = 4,(t), z = ^(t), and the surface S is giTen by a sisfle eqnatte
of the form F(x, y, «) = 0, the curre T Just defined will have eqM^
tions of the form x ^f{t), y - 4(0. « - »('). ^*>«« •XO *• • '■^
tion defined by the equation
4d2 SKEW CURVES [XI, §237
In order that T and V should have contact of order n, the infini-
tesimal 7r(t) — ij/(t) must be of order n -\- 1 with respect to t — to,
that is, we must have
T(g = 'A(^o) , 7r'(g = if^Xto) , . . • , 7r<»> (to) = V^«> (to) .
Using F(^) to denote the function considered in § 234, these equa-
tions may be written in the form
F(<«) = 0, F'(<„) = 0, ..., F<»'(<„) = 0.
These conditions may be expressed by saying that the curve and
the surface have n + 1 coincident points of intersection at their
point of contact.
If S be one of a family of surfaces which depends on n -\-l
parameters a, bj c, • • • j I, the parameters may be so chosen that S
has contact of order n with a given curve at a given point ; this
surface is called the osculating surface.
In the case of a plane there are three parameters. The equations
which determine these parameters for the osculating plane are
Af (t) + B<^ (t) i-Cil; {t)-\-D = 0,
Af(t) + B<l>'{t)-\-Cil;'(t) =0,
Af"(t) 4- B<t>'Xt) + cr(t) = 0.
It is clear that these are the same equations we found before for
the osculating plane, and that the contact is in general of the second
order. If the order of contact is higher, we must have
Af"(t) + B<f>"\t) 4- C,/.'"(0 = 0,
i.e. the osculating plane must be stationary.
237. Osculating sphere. The equation of a sphere depends on four
parameters; hence the osculating sphere will have contact of the
third order. For simplicity let us suppose that the coordinates
X, ?/, « of a point of the given curve F are expressed in terms of the
arc s of that curve. In order that a sphere whose center is (a, b, c)
and whose radius is p should have contact of the third order with
r at a given point (x, y, z) on T, we must have
F« = 0, F'(5) = 0, F"W = 0, P"(5) = 0,
where
FW = (X - ay-\-{y - by-\-{z - cy-p^
andwhereaj,y,.M6tipr8MedMfunot40Mart. Esp«MliBt Um
la8t three of the eouationi of oonditUm sod applvtiiff PWMf.
formulae, we find ''^mmfm
F'(«) = («-a)a + (y-.6)/8 + («-c)y-0.
^-<-)-T-'(^7")-'T^(i*f)-4-'a*if)
These three equations determine a, 6, and c. Bat the fini of them
represents the normal plane to the curve T at the point (», y, c) in
the running coordinates («, *, <•)» and the other two maj be deriTcd
from this one by differentiating twice with raapect to «.
the center of the osculating sphere is the point where the polar li
touches its envelope. In order to solve the three aqnattona W9 au^
reduce the last one by means of the others to the form
(X - a)a" -f Cy - b)fi" + (* - r)y" = T ^,
from which it is easy to derive the formula
Hence theradius of the osculating sphere is given by the formttla
If R is constant, the oont^r of the osculating sphere ootaeidfie wtth
the center of curvature, which agrees with the retnlt obUined la
§233.
238. Osculating straight lines. If the eqnatkms of a fiudly td
curves depend on n + 2 parameters, the parameCera may be chosen
in such a way that the resulting curve C has oontifli of ofder • with
a given surface ^^ at a point M. For the equatkm whieh exiirBaBta
that C meets S at ^f and the ft + 1 equations which expires thai
there are n + 1 coincident points of intefteeiioo ai U
n -\-2 equations for the determination of the
494 SKEW CURVES [XI, Exa.
For example, the equations of a straight line depend on four
parameters. Hence, through each point JW of a given surface 5,
there exist one or more straight lines which have contact of the
second order with the surface. In order to determine these lines,
let us take the origin at the point M, and let us suppose that the
z axis is not parallel to the tangent plane at M. Let z — F(xy y)
be the equation of the surface with respect to th^se axes. The
required line evidently passes through the origin, and its equations
are of the form
a b c
Hence the equation cp = F(ap, hp) should have a triple root p = ;
that is, we should have
c = ap -{- bq,
= a^r-i-2abs-\-bH,
where p, q, r, s, t denote the values of the first and second deriva-
tives of F{x, y) at the origin. The first of these equations expresses
that the required line lies in the tangent plane, which is evident
a priori. The second equation is a quadratic equation in the ratio
b/a, and its roots are real if s^ — rt is positive. Hence there are in
general two and only two straight lines through any point of a given
surface which have contact of the second order with that surface.
These lines will be real or imaginary according as s^ — r^ is positive
or negative. We shall meet these lines again in the following
chapter, in the study of the curvature of surfaces.
EXERCISES
1. Find, in finite form, the equations of the evolutes of the curve which
cuts the straight line generators of a right circular cone at a constant angle.
Discuss the problem.
[Licence^ Marseilles, July, 1884.]
2. Do there exist skew curves T for which the three points of intersection
of a fixed plane P with the tangent, the principal normal, and the binormal are
the vertices of an equilateral triangle ?
3. Let r be the edge of regression of a surface which is the envelope of
a one-parameter family of spheres, i.e. the envelope of the characteristic circles.
Show that the curve which is the locus of the centers of the spheres lies on
the polar surface of r. Also state and prove the converse.
4. Let r be a given skew curve, M a point on r, and a fixed point in
ipace. Through draw a line parallel to the polar line to r at 3f, and lay off
on this parallel a segment O^ equal to the radius of curvature of V at M. Show
XI, ExH.] KXKtCIBEi
that the curve r' daierlbwl bj Um potet Jiraad tte
center of cunrature of r have their tiinnoi
length equal, and their radii of eurratura aqual, aft
6. If the oieiilatiiig ipbere lo a givm tkt
show that r lies on a iphere of mdiua a, at laaei
of r ia constant and equal to a.
6. Show that the neceeeary and foffleiMit «^otKthVm Ihaft Ika
center of curvature of a helix drawn on a eyUndtr iboold bt
cylinder parallel to the first one is that the right atoikai of Uh
should be a circle or a logarithmic iplraL In tha latter CMi i
helioea lie on circular cones which hare the same azii and tba
[TissoT, NmmUm Ammmlm, Vol XI« lMi.J
7*. If two skew curres have the same prtodpal
planes of the two curves at the pohiU where they meei the
a constant angle with each other. The two potnta jnet
ters of curvature of the two curves form a sjretem of fo«r potote
monic ratio is constant. The product of the radii oi tonloa of Um two
at corresponding points is a constant.
[Paul Raaaar ; llavaHsni ; Scasu.]
8*. Let X, y, z be the rectangular coArdinatce of a poiat <m a rfww c«no P,
and 8 the arc of that curve. Then the conre r« deAned bf Ike •
Xo-fa"dM, y^sJfiTd*, s»Bj*Y''ds.
where xo, Vo, zo &re the running coOrdinatea, li oalled Iho
and the curve detined by the eqnatiooe
X = xco8tf + xosin^, F = ycos# + ir»ein#, Z » tcos#4- leilit.
where X, F, Z are the running ooflcdinatea and # ie a
a related curve. Find the orientatioQ of the
these curves, and find their radii of conratare and of
If the curvature of T is constant, the totiioB of the tnm Ft to
the related cunes are corvee of the Bertraiid tjpe (| tIS).
general equations of the latter corvee.
9. I.et r and T' be two skew corvee whkh are
A lay off infinitesimal arcs ilif and AM' from A aloag the
same diroction. Find the limiting podtloa of the Mm Mit
10. In order that a straight line rigidly
dron of a skew curve and p ee ring
describe a developable sorfaoe, that
at least unices the given skew conre to a
infinite number of etiaight Unee which hftfO tht
496 SKEW CURVES [XI, Exs.
For a curve of the Bertrand type there exist two hyperbolic paraboloids
rigidly connected to the fundamental trihedron, each of whose generators
describes a developable surface.
[CesIro, Rivista di Mathematical Vol. II, 1892, p. 155.]
11*. In order that the principal normals of a given skew curve should be the
binormals of another curve, the radii of curvature and the radii of torsion of
the first curve must satisfy a relation of the form
A
where A and B are constants.
[Mannheim, Comptes renduSf 1877.]
[The case in which a straight line through a point on a skew curve rigidly
connected with the fundamental trihedron is also the principal normal (or the
binormal) of another skew curve has been discussed by Pellet {Comptes renduSj
May, 1887), by Ceskro {Nouvelles Annates, 1888, p. 147), and by Balitrand
{Matliesis, 1894, p. 159).]
12. If the osculating plane to a skew curve T is always tangent to a fixed
sphere whose center is O, show that the plane through the tangent perpen-
dicular to the principal normal passes through O, and show that the ratio of
the radius of curvature to the radius of torsion is a linear function of the arc.
State and prove the converse theorems.
CHAPTER XII
SimFACSS
I. CURVATURE OF CURVES DRAWN ON ▲ SimPACE
239. Fundamenul formuU. Meusnier*t thtona. In oidar to ttwJT
the curvature of a surface ;it a non-singular point 1/, we f>tft H sup-
pose the surface referred to a system of rectangulAr ooOidiiialst
such that the axis of z is not pai«llel to the tangent plane at M,
If the surface is analytic, its equation may be written in the fons
where F{x, y) is developable in power series aooording to powets of
x — Xq and y — yo in the neighborhood of the point M («^, y,, s,)
(§ 194). But the arguments which we shall use do not leqoiie the
assumption that the surface should be analytic : we shall owiely
suppose that the function F{x, y), together with its first and seeond
derivatives, is continuous near the point (jr«, y,) We shaO nss
Monge's notation, pt q, r, s, t, for these deriTatiTes.
It is seen immediately from the equation of the tangent plane
that the direction cosines of the normal to the sorfaoe aie propor-
tional to 7>,.^, and — 1. If we adopt as the positive direetioo of the
normal that which makes an acute angle with the positive s
the actual direction cosines themselTOs A, m r aie girea by
formulae
-p — y I
(2)
vrT^MT* vr+^+7« vi+y+
Let C be a curve on the surface S through the point if,
the equations of this curve be given in parameter foi
functions of the parameter which represent the ooArdinates of n
point of this curve satisfy the equation (1), and heno
entials satisfy the two relations
(3) datnpdx-^-qdy,
(4) d^x =:»pd*x -k-qd^y + rds^ + 2*iUJy -f iWjf*.
497
498 SURFACES [XII, §239
The first of these equations means that the tangent to the curve C
lies in the tangent plane to the surface. In order to interpret the
second geometrically, let us express the differentials which occur in
it in terms of known geometrical quantities. If the independent
variable be the arc o- of the curve C, we shall have
dx dy _ dz _ d'^x _ a' d^y _ /3' d^z _ y'
1^^"^' d^^"' d^~^' d^~R' Io^^r' d^^~R'
where the letters a, ft, y, a', p', y', R have the same meanings as in
§ 229. Substituting these values in (4) and dividing by Vl+^^ + j^,
that equation becomes
y^ — pa^ — qP^ _ ra^ + '^sa^ + t^'^
R Vl + y + ?' "" Vl+J3^ + ^^
or, by (2),
Xa^ + /M)8' + vy' _^ ra^ + 2sap -f t/S^
R ~ vrr^M^'
But the numerator Xa' + fip' + vy' is nothing but the cosine of the
angle 6 included between the principal normal to C and the positive
direction of the normal to the surface ; hence the preceding formule
may be written in the form
cos e ra^ + 2safi -f tfi^
(5)
R Vl+y4-?'
This formula is exactly equivalent to the formula (4); hence it
contains all the information we can discover concerning the curva-
ture of curves drawn on the surface. Since R and Vl -\-p^ -\- q^
are both essentially positive, cos 6 and ra^ + 2sa/3 -f tjS'^ have the same
sign, i.e. the sign of the latter quantity shows whether is acute or
obtuse. In the first place, let us consider all the curves on the sur-
face S through the point M which have the same osculating plane
(which shall be other than the tangent plane) at the point M. All
these curves have the same tangent, namely the intersection of the
osculating plane with the tangent plane to the surface. The direc-
tion cosines a, ft, y therefore coincide for all these curves. Again,
the principal normal to any of these curves coincides with one of
the two directions which can be selected upon the perpendicular to the
tangent line in the osculating plane. Let w be the angle which the
normal to the surface makes with one of these directions ; then we
shall have 6 = 0) or = tt — <d. But the sign of 7-a'^ -h 2saft -f tfi^
shows whether the angle is acute or obtuse ; hence the positive
XII, J2»] CURVK.s <>:> A r>LKrAC£
direction of the principal normal i« ihe aame for all
Since $ is also the same for all the curvM, the radiua of
R is the same for them all ; that is to tay, dY/ rJU rmrwm 0m ik§ «ww
face through the poiiU M which have ths mm
the same center of curvature.
It follows that we need only study the ounralim of the
sections of the siirface. First let us study tht TtriMMMi of Ite
curvature of the sections of the surface by planes whieh all mm
through the same tangent MT, We may suppoee, witiiovl koo of
generality, that ra^ + 28a p -f t/J* > 0, for a ohaoge in the dtioette
of the z axis is sufficient to change the signs of r, $, and t. For all
these plane sections we shall have, therefore, ooi^>0, and tlM
angle 6 is acute. If /?| be the radius of ourrateva of the aortinii
by the normal plane through 3/r, since the oorreapoodiaf aafle #
is zero, we shall have
Comparing this formula with equation (5), which giToe the ladiaa
of curvature of anv obli(iue seotioOy we find
/AN ^ _ooe^
(^> J, '"IT'
or R = Ri cos 6, which shows that the eemier ^ m nmimr e •/ mtf
obliqm section U the prcjeetum of the emter of rurraimre
normal section through the 9ame tawgemi Um$, Tbta it llett»titrr »
theorem. '
The preceding theorem reduces the study of the eafralBio of
oblique sections to the study of the ourrature of nonuJ ioelkMH.
We shall discuss directly the results obtained by Bokr. Flwl Isl
us remark that the formula (6) will appear in two differsBl fonM
for a normal section according as nr* 4* 2sa/l + I/I* is podtivo or
negative. In order to avoid the inoooTOoioDOS of eanyinf %hmB
two signs, we shall agree to affix the sign -»- or the sign - to Iho
rjulius of curvature /? of a normal soetion aeoordinc as the dlrsette
from M to the center of cunrature of the ssetiaii is tho sum as or
opposite to the positive direction of the normal to the sorlWts.
With this convention, R is given in either ease by tho formula
m 1 ra»^2eafi + tf
600 SURFACES [XII, § 239
which shows without ambiguity the direction in which the center
of curvature lies.
From (7) it is easy to determine the position of the surface with
respect to its tangent plane near the point of tangency. For if
s^ - rt <0, the quadratic form ra'^ -\- 2sap + tfi^ keeps the same
sign — the sign of r and of ^ — as the normal plane turns around
the normal; hence all the normal sections have their centers of
curvature on the same side of the tangent plane, and therefore all
lie on the same side of that plane : the surface is said to be convex
at such a point, and the point is called an elliptic point. On the
contrary, if s^ — rt > 0, the form ra^ -|- 2sap + tfi^ vanishes for two
particular positions of the normal plane, and the corresponding
normal sections have, in general, a point of inflection. When the
normal plane lies in one of the dihedral angles formed by these two
planes, R is positive, and the corresponding section lies above the tan-
gent plane ; when the normal plane lies in the other dihedral angle,
R is negative, and the section lies below the tangent plane. Hence
in this case the surface crosses its tangent plane at the point of
tangency. Such a point is called a hyperbolic point. Finally, if
s^ — rt = 0, all the normal sections lie on the same side of the tan-
gent plane near the point of tangency except that one for which
the radius of curvature is infinite. The latter section usually
crosses the tangent plane. Such a point is called a parabolic point.
It is easy to verify these results by a direct study of the differ-
ence n, — z — »' of the values of z for a point on the surface and for
the point on the tangent plane at M which projects into the same
point (a;, ?/) on the xy plane. For we have
z' z=p{x - x^ ^ q{y - y^,
whence, for the point of tangency {x^, ?/o)>
dx "'
1^ = 0.
dy
U
;5 = ^
dxdy"^'
and
dx
It follows that if «^ — r^ < 0, w is a maximum or a minimum at M
(§ 56), and since n vanishes at M, it has the same sign for all other
points in the neighborhood. On the other hand, if 5* — r< > 0, u
has neither a maximum nor a minimum at AT, and hence it changes
sign in any neighborhood of M.
240. Ettler*! theoremt. The IndioitHx. in onler lo tlody lb« w^
tioii uf the radius of curvaturu of a nomud Molioo, \h m t^km tkt
point M as the origin and the tangent pbuie at AT m Um xy f^^m
With such a system of axes we shall haTO j» » ^ « o, aad Um
formula (7) beoomee
(8) ii ~ ** °^**^ + 2f cos 4 sin ^ + < tinV*
where <^ is the angle which the trace of the normal plaae
with the positive x axis. Equating the derivatiTe of Um
inembcr to zero, we find that the pointa at which H maj In a
mum or a minimum stand at right angles. The following geomei^
rical picture is a convenient means of visualizing the TariaticMi of M,
Let us lay off, on the line of intersection of the normal plane wtlk
the xf/ plane, from the origin, a length Om equal numertcallj to Um
square root of the absolute value of the corresponding radios of
vature. The point m will describe a corrs, which gires an
neous picture of the variation of the radius of curratniv. This carra
is called the indiccUrix, Let us examine the three possibls oassa.
1) s^ — rt <0. In this case the radius A has a coostaat sign, whleb
we shall suppose positive. The coordinates of m are i « VS cos 4
and i; = V^ sin <t> ; hence the equation of the imJitairiM is
(9) re-^2iif, + t^^l,
which is the equation of an ellipse whose center is the orifia. It is
clear that Ji is at a maximum for the section made bf the aocaMl
plane through the major axis of this ellipse, and at a miaimam for
the normal plane through the minor axis. The sections omde bf two
planes which are equally inclined to the two axes eridentljr hare the
same curvature. The two sections whose planes pass through the
axes of the indicatrix are called the primeipai monmmi m tt im$ t aad
the corresponding radii of curvature are called the ptimtipmi mHi ^
curvature. If the axes of the indicatrix are taken for the axas of a
and y, we shall have # = 0, and the formula (8) heooHMt
4 = rcosV + <«n*4.
n
With these axes the principal radii of cvrratare JI, aad if,
to <^ = and «^ = 7r/2, respectively; hence l/i», - r, !/£, - I,
1 cos*4 ^ «P*»
502 SURFACES [XII, § 240
2) s^ — rt> 0. The normal sections which correspond to the
values of <^ which satisfy the equation
r cos^<f> + 2« cos <^ sin <^ + ^ sin^</) =
have infinite radii of curvature. Let ZjOZi and L^OZg be the inter-
sections of these two planes with the xij plane. When the trace of
the normal plane lies in the angle ZjOZg, for example, the radius
of curvature is positive. Hence the corresponding portion of the
indicatrix is represented by the equation
where i and r) are, as in the previous case, the coordinates of the
point m. This is an hyperbola whose asymptotes are the lines
L[OLi and L^OL^. When the trace of the normal plane lies in the
other angle L^OLu R is negative, and the coordinates of m are
I = V— R cos <^ , ly = V — i2 sin <^ .
Hence the corresponding portion of the indicatrix is the hyperbola
re + 2s^'q-\-trf = -l,
which is conjugate to the preceding hyperbola. These two hyper-
bolas together form a picture of the variation of the radius of curva-
ture in this case. If the axes of the hyperbolas be taken as the
X and y axes, the formula (8) may be written in the form (10), as in
the previous case, where now, however, the principal radii of curva-
ture Rx and R^ have opposite signs.
3) s^ — rt = 0. In this case the radius of curvature R has a
fixed sign, which we shall suppose positive. The indicatrix is still
represented by the equation (9), but, since its center is at the origin
and it is of the parabolic type, it must be composed of two parallel
straight lines. If the axis of y be taken parallel to these lines, we
shall have s = 0, t = Oj and the general formula (8) becomes
or
1 _ cos'<^
R" Ri '
This case may also be considered to be a limiting case of either of
the preceding, and the formula just found may be thought of as the
limiting case of (10), when R^ becomes infinite.
XII, 5 241] CURVES ON A 8URPACK
Euler'sforaolsmaybettUblUMd wiUuMiftwli«ltolanMk<«|. T^kkm
the point M of tbe given mutMM m Um origiii Ami lb* Tinjiiii |iiaM m iJiIiw
plane, tbe ezpausion of 1 1^ Taylor*! mHm mi^ bt wHttia lo tW fons
l.S ^"'*
where the terms not written down are of order grMier 'kft t««. In nidn
tu tiiid the radii of curvature of the notion made bj n pUne y ■ a taa a, ve
may introduce the transformntion
X = s'ooe^ - y'iln^, y = z'lln^ + V'cm^,
and then set / = 0. Tble gives the expansion of z in powets of x',
^ _ rcos«^ -t- >ssin»ooa^ ^. f sin«»
1.2 r'^ . .
which, by § 214, leads to the formuhi (8)
Notes. The section of the surface by iu langeni plane Is given by the eqoalloa
= rx« + 2sa5y + fyt + #,(x, r) + • • •.
and has a double point at tbe origin. Tbe two tangenU at this potol aie the
asymptotic tangents. More generally, if two surfaces 8 and 8% an holh tM^Hl
at the origin to the xy plane, the projection of their curve of InlenscUea on iha
xy plane is given by the equation
= (r - ri)x« + 2(« - $i)xy -^t - <,,yi ^ . . .,
where ri, Si, ti have the same meaning for the Aurfaos S% that r, a, f have
for S. Tlie nature of the double point depends upon the dfn ol the expceaten
(s — «i)> — (r — ri)(t — fi). If this exprenlon ia zrro, the curw uf int Jif— r U ca
has, in general, a cusp at the origin.
To recapitulate, there exist on any surface four reoiarkmbU pon>
tions for the tangeut at any point : two peqwndioular tanceota for
which the corresponding radii of curvature have a maxiinttm or a
minimum, and two so-<ralled asympioiiej or prtnripal* tangvnta, for
which the corresponding radii of curvature are infinite. The latter ara
to be found by equating the trinomial ra*4- 2sa/J+ f/J* to aaro (| W^
We proceed to show how to find tlie principal nonaal a ec ii oi n aad
the principal radii of curvature for any system of rectangular
241. Principal radii of curvature. There are in geoerml two
normal sections wliose radii of curvature are equal to any fives
value of R. The only exception is the caee in whieh the five©
value of iJ is one of the principal radii of ourvalnre, in which ease
• The reader should distinguish sharply the dl wetto as of
(the asymptotes of tbe Indlcatrix) and Ihs dlrwjtloas of ^
(thetuccvof the Indicatrlx). To avoid
tangent.— Ikkhs.
604 SURFACES [XII, §241
only the conesponding principal section has the assigned radius
of curvature. To determine the normal sections whose radius of
curvature is a given number R, we may determine the values of
a, p, y by the three equations
Vl 4- »2 -f </^
^ = ra'' + 2safi + tjS^ y=pa-\-q/3, a* + ^^ + y' =1 .
It is easy to derive from these the following homogeneous combiner
tion of degree zero in a and y8 :
nU Vl + j>^ + g^ ^ ra^ + 2sal3 + t/S
^ ^ R a' + ^ + {pa^qP)
2
It follows that the ratio p/a is given by the equation
a\l + p^ - tD) + 2ap{pq - sD) -^ ^"(1+ q^ - tD) = 0,
where R — i) Vl -\-jp^ -\- (f. If this equation has a double root, that
root satisfies each of the equations formed by setting the two first
derivatives of the left-hand side with respect to a and ^ equal to
zero:
(12)
a{pq-sD)^^{^L-^q^-tD) = 0.
Eliminating a and fi and replacing 2) by its value, we obtain an
equation for the principal radii of curvature :
(13) j
(14)
{rt-s')R''--\l\^f^q\{\-^p'^t-\-(^-\-q^)r-2pqs'\R
+ (l+i>' + ?y = 0.
On the other hand, eliminating D from the equations (12), we obtain
an equation of the second degree which determines the lines of inter-
section of the tangent plane with the principal normal sections :
a\{l-\- f)s - pqr\
Jrafi\^{\-\-f)t-{\^q^)r-\^f^lpqt-Q.^q^)s-\=.{).
From the very nature of the problem the roots of the equations (13)
and (14) will surely be real. It is easy to verify this fact directly.
In order that the equation for R should have equal roots, it is
necessary that the indicatrix should be a circle, in which case all
the normal sections will have the same radius of curvature. Hence
the second member of (11) must be independent of the ratio j8/a,
which necessitates the equations
^^'"'^ 1 + y pq l + ?»*
XII. §241] CURVES ON A 8UKFACB 501
The points which satisfy theM equations ar« exiled MwtWw, jU
such points the equation (14) roduoat to an idaatilj, alaM •tttj
diameter of a circle ia also an axis of ajnuBOlrj.
It is often possible to determine the prinetpal
from certain geometrical considerationa. For *nffttttim. If a
S has a plane of symmetry through a point M od tba tarfaM* H b
clear that the line of intersection of that plana with the
plane at ^f is a line of symmetry of the indioatrix ;
tion by the plane of symmetry is one of the principal
example, on a surface of revoluticm the meridian through aay podit
is one of the principal normal sections ; it is evident that tlia
of the other principal normal sectioo passes through tha
the surface and the tangent to the circular parallel at the polat.
But we know the center of curvature of one of the obliqi
through this tangent line, namely that of the oiroular paralUI U
It follows from Meusnier's theorem that the oentar of ciinratiira of
the second principal section is the point whara tha nonaal to tkm
surface meets the axis of revolution.
At any point of a developable surface, «* — H = 0, and tha iadiea-
trix is a pair of parallel straight lines. Ona of tha prindpal 8ia>
tions coincides with the generator, and tha corresponding radios of
curvature is infinite. The plane of the second principal saotion is
perpendicular to the generator. All the points of a devalopabla
surface are parabolic, and, conversely, thasa are the only siufa aaa
which have that property (§ 222).
If a non-developable surface is convex at certain points, while oIImt
points of the surface are hyperbolic, there is usually a line of pam>
bolic points which separates the region where j* - r< is positive froM
the region where the same quantity is negative. For eia mp K os thm
anchor ring, these parabolic lines are the aztNOM eirenlar p f l M i
In general there are on any convex sorfMe only a iafteawiir efaaiMtaa.
We proceed to show that the only real MilBee lor wUek trm j p«ii > le m
umbilic is the sphere. Let X, m. ' be the direetioa edilB« of lbs aoiaal le Ike
surface. Differentiating (2), we find the formote
d\_ pq$'-{l + q^r 9X ^ pqi-il-^ ^ »
to" (l + pi + g«)« ' *r {X^l^-^^
"■"""• g... g... s-g-
606 SURFACES [XII, § 242
The first equation shows that X is independent of y, the second that fj. is inde-
pendent of X ; hence the common value of d\/dx, dfx/dy is independent of both
X and y, i.e. it is a constant, say 1/a. This fact leads to the equations
. _X-Xo
A = »
a
_y -yo _ Va2 - (X - Xo)2 -
a ' a
-{y-
-J/o)^
P =
X X — Xo
V y/cC^ -{X- Xo)2 - (y - yo)^
Q =
_ A* _ y-yo
y Va2 - (X - Xo)2 - (y - yo)2
>, integrating,
the value of z is found to be
z = zo + v a'-* - (ic - ^oY -{y - yo)'^ ,
which is the equation of a sphere. It is evident that if d\/dz = dfi/dy = 0, the
surface is a plane. But the equations (15) also have an infinite number of
imaginary solutions which satisfy the relation I + p^ + q^ = 0, as we can see by
differentiating this equation with respect to x and with respect to y.
II. ASYMPTOTIC LINES CONJUGATE LINES
242. Definition and properties of asymptotic lines. At every hyper-
bolic point of a surface there are two tangents for which the corre-
sponding normal sections have infinite radii of curvature, namely
the asymptotes of the indicatrix. The curves on the given surface
which are tangent at each of their points to one of these asymptotic
directions are called asymptotic lines. If a point moves along any
curve on a surface, the differentials dx, dy, dz are proportional to
the direction cosines of the tangent. For an asymptotic tangent
ra^ 4- 2sap -\-tp'^ = 0] hence the differentials dx and dy at any point
of an asymptotic line must satisfy the relation
(16) rdx^-\-2sdxdy-{-tdy^ = 0.
If the equation of the surface be given in the form z = F(x, y), and
we substitute for r, s, and t their values as functions of x and y,
this equation may be solved for dy/dx, and we shall obtain the two
solutions
(^^) £=*'(^'j')' 2 =*'(*' 2')-
We shall see later that each of these equations has an infinite num-
ber of solutions, and that every pair of values (xq, ?/o) determines
in general one and only one solution. It follows that there pass
through every point of the surface, in general, two and only two
Xll,ja42] ASYMPTOTIC LINES rovJi^AiK LOIBI ^
asymptotic lines : all these linee togeUier torm * doohto
lines upou the surfaoe.
Again, the asymptotio lines may be defined wtthom Iht me el
any metrical relation : ths a»ympt4ttie Umm e« « „|,y,Dg ^^ ^j^^^
curves for which the oseuloHng plans alway§ rtimiim wkk Us 9^
gent plane to the tur/aee. For the neoeaaary and siifleieoi eottdilMMi
that the osculating plane should ooinoide with the teBgwt dImm to
the surface is that the equationa
dz —pdx — qdy = (i^ d*z — pd*x — ytPy ^
should be satisfied simultaneously (see f 215). The fif«| of Umm
equations is satisfied by any curve which liee oo the torfM. Dif*
ferentiating it, we obtain the equation
d*z—pd*x — qiPy -dpdx -dqdymO,
which shows that the second of the preoeding eqaationfl may be
replaced by the following relation between the firit diffeiwitiaU :
(18) dpdx-^dqdy^O,
an equation which coincides with (16). Motmrm it ia Mij to
explain why the two definitions are equivalent Sinoe the radios of
curvature of the normal section which is tangent to an asjmptoii
of the indicatrix is infinite, the radius of curmtore of Um Mymp*
totic line will also be infinite, by Meusnier^i theorem, at leaat unUi
the osculating plane is perpendicular to the normal plane, in whkk
case Meusnier's theorem becomes illusory. Henee the
plane to an asymptotic line must coincide with the taag
at least unless the radius of curvature is infinite ; but if thia
true, the line would be a straight line and ita oeru l ating
would be indeterminate. It follows from this property thai aaj
projective transformation carries the asymptotio linea iato MX^P-
totic lines. It is evident also that the diffeteiitial e^iakte it of
the same form whether the axes are rectangular or ohUqM, for the
equation of the osculating plane remains of the smm Hnu
It is clear that the asymptotic lines exiat only i» «••• ^ potato of
the surface are hyperboUo. But when the tiirlMi It ato^jlta the
differential equation (16) always has an infinite noaber of eol»-
tious, real or imaginary, whether i" - »f is poailive or neflftivm. As a
generalization we shall say that any convex twlMt poitotoM two ey^
tems of imaginary asymptotic lines. Thus the aaynptode liato «f n
unparted hyperboloid are the two tyttma of roeliliiiear
508 SURFACES [XII, § 24;{
For an ellipsoid or a sphere these generators are imaginary, but
they satisfy the differential equation for the asymptotic lines.
Example. Let us try to find the asymptotic lines of the surface
z = x"*y*^.
In this example we have
r = m(m — l)a;'"-2y«, 8 = mnx"^-^y'*-^j < = n(n — l)x™y»-2^
and the differential equation (16) may be written in the form
mlm - 1) ( ^ I + 2mn( ^ ) + n(n - 1) = 0.
\xdy/ \xdyj
This equation may be solved as a quadratic in {ydx)/{xdy). Let hi and ^ be
the solutions. Then the two families of asymptotic lines are the curves which
project, on the xy plane, into the curves
yfii=CiX, yf'i=C2X.
243. Differential equation in parameter form. Let the equations of
the surface be given in terms of two parameters u and v :
(19) x= f(u, v), y = 4>{u, v), z = il/(uy v) .
Using the second definition of asymptotic lines, let us write the
equation of the tangent plane in the form
(20) A(X-x)-\-B(Y-y)-hC(Z-z) = 0,
where A, B, and C satisfy the equations
(21)
cu cu ou
cv cv cv
which are the equations for A, B, and C found in § 39. Since the
osculating plane of an asymptotic line is the same as this tangent
plane, these same coefficients must satisfy the equations
Adx +Bdy -\- C dz =0,
Ad^x-\-Bd^yhCd^z=0.
The first of these equations, as above, is satisfied identically. Differ-
entiating it, we see that the second may be replaced by the equation
(22) dAdx + dBdy-\-dCdz = 0,
which is the required differential equation. If, for example, we
set C = — 1 in the equations (21), A and B are equal, respectively,
to the partial derivatives p and q oi z with respect to x and y, and
the equation (22) coincides with (18).
XU,5244] ASYMPTOTIC LINES CONJUGATE UXES
£xamp<ef. Ata&«zaaptol«laieoBil<tortJMooDo4dt«#<y/k). VMrnm^
tion 18 equivalent to the (qntom SBii,ysMp,fl. ^(t), m4 y^ tqMlliM Oil
beconte
These eqaationn are ■atiifled IfweaetCsr^n,^-. »^'(,). j| ;= ^-i^ . *^-^-
the equation (22) takee the form
One solution of this equation i« o = oonat, whieh givM ih«
tors. Dividing by do, the remaining equation ia
#^(e)d» _»dM
whence the second system of asymptotic lines are the
defined by the equation u« = Jir^'(o), which
curves
-'••©
Again, consider the surfaces discnased by Jamst, whose sqoalk» may be
written in the form
Taking the independent variables t and u = p/x^ the
the asymptotic lines may be written in the
from which each of the systems of aqrmpCoUc lines may be fooad by a slagls
quadrature.
A helicoid is a surface defined by equations of the form
xspcosM, y = psinw /(^)-fA».
The reader may show that the differential eqaation of tbe aqravtotio iMi ll
p/"{p)df^ - 2h<Utdp + f^r{p)d»^ « 0,
from which u may be found by a single qoadralara.
244. Asymptotic Unee on a mled niifiot. Eliminaiuig A,B,Ukdr
between tbe equations (21) and tbe eqnatioQ
we find the general differential eqoatioQ of the mfymptolie Mum :
dl d± di
du du ^
(23) ^ ?* ?*
dv dv hr
d^x d*y €Pm
«0.
510
SURFACES
[Xn, §244
This equation does not contain the second differentials <Pu and d'^v,
for we have
d'^x = ^-fd^u^% dH + ^(ft^2 + 2 ^ dudv\-^^^dv^
and analogous expressions for d!^y and d^z. Subtracting from the
third row of the determinant (23) the first row multiplied by d^u
and the second row multiplied by d'^v, the differential equation
becomes
df_ d_^ d_^
du du du
dv dv dv
^'/ , ,
■^dudv + Y^dv^
cu cv cv^
= 0.
Developing this determinant with respect to the elements of the
first row and arranging with respect to du and dv, the equation
may be written in the form
(24) D du^ + 2D'dudv-h D"dv^ = 0,
where D, D', and D" denote the three determinants
(25)
D =
dx
du
dx
dv
d^
du^
dy^
du
djy^
dv
a^
du''
d_z
du
d_z
do
d^
du^
D" =
dx
du
hi
du
dz
du
D
1
dx
dv
dy
dv
dz
dv
d^x
d'y
d^z
dudv
dudv
dudv
dx
d^j
dz
du
du
du
dx
dy
dz
da
dv
dv
'
d^x
d'y
d^z
dv^
dv'
du'
As an application let us consider a ruled surface, that is, a surface
whose equations are of the form
x = Xq-^ auj y = yQ + P^h Z = ZQ-\-yU,
where Xq, ?/o, Zq, a, )3, y are all functions of a second variable param-
eter V. If we set w = 0, the point {xq, yo, Zq) describes a certain
curve r which lies on the surface. On the other hand, if we set
V = const, and let u vary, the point (x, y, z) will describe a straight-
XII, J 9W] ASYMPTOTIC LINES OOXJUQATl Uns 611
line generaU)r of the ruled surface, ftnd the Tmltae at m U uv nam
of the line will be proportional to the dietouiee hrtirw tW mIM
(Xf y, z) and the point (ar«, y«, ««) at which the g ww iaiw HMeto tiM
curve r. It is evident from the formulfi* (25) that #> ■> 0, tlMt V
is independent of u, awA tliai />" Im a polynomial of fhf
degree in «:
Since <fr is a factor of (24), one system of asymplotie lii
of the rectilinear generators v ss const DiTidinff by dm^ the
ing differential etjuation for the other system of asympCoik Uaea ia
of the form
(^^^ ~ -f Lu« + Afit + JV « 0,
where L, 3/, and N are functions of the single Taiiable «. As eqim*
tion of this type possesses certain remarkable propartiei^ wkiell «•
shall study later. For example, we shall aee thai lA# ««A«rHeafr
ratio of any four solutions is a constanL It followa thai tlie anlMr-
monic ratio of the four points in which a generator meets a^y fov
asymptotic lines of the other system is the same for all
which enables us to discover all the aaymptotic Itnet of the
system whenever any three of them are known. We shall also
see that whenever one or two integrals of the equatkio (M) aie
known, all the rest can be found by two qnadratares or by a aiafle
quadrature. Thus, if all the generators meet a fixed stmiflil Uas^
that line' will be an asymptotic line of the aeoood fyftam, aad all
the others can be found by two quadraturea. If tlie snrCaee pea-
sesses two such rectilinear directrices, we should know two
totic lines of the second system, and it would appear that
quadrature would be required to find all the othata. Bol w« mm
obtain a more complete result. For if a tnrface poaaeaaaa Isfo
rectilinear directrices, a projective transformaUoo can be foasd
which will carry one of them to infinity and tranafonn the avrfaee
into a conoid ; but we saw in f 243 thai the aaymplolk Iteea m a
conoid could be found without a single quadratura.
245. Conjugate lines. Any two conjugate diamelara of the todk s
trix at a point of a given surface S are called es^afafe
To every tangent to the surface there eocreipQwIa a <
tangent, which coincides with the fif*i when and only vhen
t
512 SURFACES [Xii,§245
tangent is an asymptotic tangent. Let z = F{x^ y) be the equation of
the surface 5, and let m and m) be the slopes of the projections of
two conjugate tangents on the xy plane. These projections on the
xy plane must be harmonic conjugates with respect to the projec-
tions of the two asymptotic tangents at the same point of the sur-
face. But the slopes of the projections of the asymptotic tangents
satisfy the equation
r4-2s/Lt + ^/x2 = 0.
In order that the projections of the conjugate tangents should be
harmonic conjugates with respect to the projections of the asymp-
totic tangents, it is necessary and sufficient that we should have
(27) r -j- s(??i + m') + tmrn) = 0.
If C be a curve on the surface S^ the envelope of the tangent
plane to 5 at points along this curve is a developable surface which
is tangent to S all along C. At every 'point M of C the generator of
this developable is the conjugate tangent to the tangent to C. Along
C, X, y, z, p, and q are functions of a single independent variable a.
The generator of the developable is defined by the two equations
Z-z-p{X-x)-q{Y-y) = 0,
— dz -\- p dx -{- q dy — dp(X — x) — dq{Y — y) = 0,
the last of which reduces to
Y—y_ dp _ rdx + sdy
X — X dq sdx-\- tdy
Let m be the slope of the projection of the tangent to C and m' the
slope of the projection of the generator. Then we shall have
dy Y — y ,
and the preceding equation reduces to the form (27), which proves
the theorem stated above.
Two one-parameter families of curves on a surface are said to
form a conjugate network if the tangents to the two curves of the
two families which pass through any point are conjugate tangents
at that point. It is evident that there are an infinite number of
conjugate networks on any surface, for the first family may be
assigried arbitrarily, the second family then being determined by a
differential equation of the first order.
xn,f24aj ASYMPTOTIC LUrE8
CONJUGATE LINES $%$
Given a Burfaee njpnmniMd by rpiiHrm ol tW j
conditions under which Um eortm u » ooml aad t ■
network. If we more Along the eorre t m eoom., i
ungent plaru* is repreeented by the two ^ ^ ef tg ne
^(X-x) + B(r-y) + C(£-i)«0.
In order that thia atraight Une aboald oolneide with the laoctat to ite tmm
u = const., whoee direction coainea are proportional to Ik/Ti, ly/l«, H/H, k
is necesaary and aufficieot that we ahoald have
DifferenUating the drat of theoe eqoationa with regard to «. we aee thai the
second may be replaced by the equation
a^ + bI^ + c^
dudv dud* and*
0,
and finally the elimination of A^ B, and C between the
leads to the necessary and aolBoieiit eondition
0.
dudv tucv iMd9
This condition is eqaivalent to Mylag that r, y. t are
differential equation of the form
(SI) and (V)
du
9V
du
dM
du
dz
dv
9V
fv
dM
d9
d^x
r^t
ofa
(29)
where M and N are arbitrary funotiooa of u and e. It ioOoWB thai the kaowl*
edge of three distinct integrala of an eqnatloa of ihta fonn le
determine the equations of a forfaoe whleh li l ii wi e d to a
For example, if we set If = iV = 0,
the sum of a function of u am
equations are of the form
of the
a funetlon of t : henee, ob anv aarfMO
(30) X =/<«)+/,(•),
the curves (u) and (v) form a ooQjogale net'
Surfaces of the type (80)
may be described in two diflbrent ways by
translation such that one of iu poinu
IT = ^11)4- ♦li*), « « tH«) + H{9)^
514 SURFACES [XII, §246
let3fo, Mi^ Mi, Mhe four points of the surface which correspond, respectively,
to the four sets of values (wo, Uo), (u, Uo)> (wo, v), (u, v) of the parameters u and u.
By (30) these four points are the vertices of a plane parallelogram. If Vq is fixed
and u allowed to vary, the point Mi will describe a curve F on the surface ; like-
wise, if Mo is kept fixed and v is allowed to vary, the point M2 will describe
another curve T' on the surface. It follows that we may generate the surface by
giving r a motion of translation which causes the point M^ to describe T', or by
giving r' a motion of translation which causes the point Mi to describe T. It is
evident from this method of generation that the two families of curves (it) and (v)
are conjugate. For example, the tangents to the different positions of T' at the
various points of T form a cylinder tangent to the surface along T ; hence the
tangents to the two curves at any point are conjugate tangents.
III. LINES OF CURVATURE
246. Definition and properties of lines of curvature. A curve on a
given surface S is called, a line of curvature if the normals to the
surface along that curve form a developable surface. If z =f(x, y)
is the equation of the surface referred to a system of rectangular
axes, the equations of the normal to the surface are
.3^. iX = -pZ + (x+pz),
^ ^ \Y=-qZ -\-{y+qz).
The necessary and sufficient condition that this line should describe
a developable surface is that the two equations
(32) |_
Zdp-{- d(x H- pz) = 0,
Zdq-{- d(y -\-qz)=:0
should have a solution in terms of Z (§ 223), that is, that we
should have
d(x + pz) _ d(y H- q^)
dp dq
or, more simply,
dx -\- pdz _ dy •\- qdz
dp dq
Again, replacing dz^ dp^ and dq by their values, this equation may
be written in the form
{\ + p'^)dx+pqdy ^ pqdx + {\-\-q^)dy
^ ^ rdx -\- sdy sdx -\- tdy
This equation possesses two solutions in dy/dx which are always
real and unequal if the surface is real, except at an umbilic. For,
if we replace dx and dy by a and p, respectively, the preceding
XII. $246] LINES OP CURVATUU 615
equation coincides with the equation found above [(U), 1 241] tm
the determination of the linea of inteneeHoo of Um sriMip*! sonBttl
sections with the tangent plane. U follows tbtl tbo timun to U^
lines of curvature through any point ooincido with th» MMm of tW
indicatrix. We shall see in the studj of diibroDtaal oqualmt IImI
there is one and only one line of curratoro throufh overr WH^
singular point of a surface tangent to eooh ooo of iho nm of tko
indicatrix at that point, exoept at ao umbilic. Tbiw Umm aio
always real if the surface ia real, and the network wkieli Ihij 1mm
is at once orthogonal and conjugate, — a charactericUo pfopottj.
Example. Let us determine the linee of oonrataiv ai Um grfti ^r rt irfct g ■
xy/a. In this example
u a
and the differential etiaation (38) Is
(a« + ya)dx« = (a« + x«)dy« or — ~ x- O
If we take the positive sign for both radJcalt, the
(x + VS^T^Xy + V?nn?) « c.
which gives one system of lines of ourratiira. If wi ni
(34) XzzxVFT^ + yViiT?.
the equation of this system may be writtaa in the lerai
X 4. VX« + <i« = C
by virtue of the identity
(x Vu^TTfl -f y Vx« + aO* + o* = [«r + V(^ -»• ^d^ + ^f
It follows that the projections of the lioet of enrvataie of tMi
represented by the equation (34), where X Is an arbitrary
sliown in the same manner that the proj«etioos of the lines of
other system are represented by the equatloo
(36) X VFTo* - y VPTi? ■ #.
From the equation xy = ac of the gitrwi paraboloid, the n e aU— IS4) aM
(35) may be written in the form
Vx« + «« + Vy« + «« = C, Vi^ + H-Vy« + «»«C.
But the expressions vV + f« and Vf^ + i* npTHMl,
tances of the point (x, y, s) from the axes of a aad y. li
of curvature on the paraboloid art tkom cufWi /br wkkk Of mm er
(ifthediManeetiUfamvpoiMupomthmfnmtktmmnfMnd^Um
51G SURFACES [xn, § 247
247. Evolute of a surface. Let C be a line of curvature on a sur-
face S. As a poiut M describes the curve C, the normal MN to the
surface remains tangent to a curve r. Let (X, Y, Z) be the coor-
dinates of the point A at which MN is tangent to r. The ordinate
Z is given by either of the equations (32), which reduce to a single
equation since C is a line of curvature. The equations (32) may
be written in the form
{\-\-p^dx-\-pqdy pq dx-\-{l-\- q^) dy
^ — ^ — ' ~~' •
rdx -\- sdy sdx -\- tdy
Multiplying each term of the first fraction by dx, each term of the
second by dy, and then taking the proportion by composition, we
find
^ ^ ^ <^x^ + dy"" + {pdx -^qdyY
r dx^ -i- 2sdxdy -^ t dy^
Again, since dx, dy, and dz are proportional to the direction cosines
a, ^, y of the tangent, this equation may be written in the form
Z -z
a' + ^^-h (pec 4- qPY ^ 1
Comparing this formula with (7), which gives the radius of curva-
ture R of the normal section tangent to the line of curvature, with
the proper sign, we see that it is equivalent to the equation
(36) Z-z= ^
where v is the cosine of the acute angle between the z axis and the
positive direction of the normal. But « + /?v is exactly the value
of Z for the center of curvature of the normal section under con-
sideration. It follows that the point of tangency A of the normal
MN to its eniielope V coincides with the center of curvature of the
principal normal section tangent to C at M. Hence the curve F is
the locus of these centers of curvature. If we consider all the lines
of curvature of the system to which C belongs, the locus of the cor-
responding curves F is a surface 2 to which every normal to the
given surface S is tangent. For the normal MN, for example, is
tangent at A to the curve F which lies on 2.
The other line of curvature C through M cuts C at right angles.
The normal to S along C is itself always tangent to a curve F'
which is the locus of the centers of curvature of the normal sections
xu,§a«]
LINES OF CURVATUU
•ill
tangent to C. The iocus oi this ounre P for all tii» Umm oC
ture of the system to whiub C belongs is a surfaoe T to wkkk all
the normals to s are Ungent The two surfaiwi 1 and I* aiv aol
usually analytically distinct, but form two nappsa of tiM mam av-
face, which is then represented by an irredndUa aqnatkm.
The normal Ai/i to S is tangent to each of these nappes S aad 1*
at tlie two principal centers of curvature A and A* of tiM siiifaai B
at the point Af. It is easy to find the tangent
planes to the two nappes at the points A and A*
(Fig. 51). As the point .U describes the curro
C, the normal MN describes the developable
surface D whose edge of regression is T; at
the same time the point A ' where AfX touches
2' describes a curve y' distinct from T, since
the straight line MN cannot remain tangent to
twu distinct curves T and V. The developable
D iind the surface X are tangent at ^4'; hence
the tangent plane to 2' at i4' is tangent to D
all along AfN. It follows that it is the plane
NMT, which passes through the tangent to C.
Similarly, it is evident that the tangent plane
to 2 at A is the plane NAfT' through the tan-
gent to the other line of curvature C*.
The two planes NAfT and NAfT' stand at right
leads to the following important conception. Let a normal OM be
dropped from any point O in space on the surface S, aad lal A and
il' be the principal centers of curvature of S on this BonaaL The
tangent planes to S and 2' at ^4 and A', respecltvalj» are pa r paad J B'
ular. Since each of these planes passes through Um gives poial C^ it
is clear that the two nappes of the 9»ohiie of amy smrfaet S. tk m r r^
from any point O in tpatty appear to cut eaek •CA^r at rifkt amftm .
The converse of this proposition will be proved later.
W^ St
This fact
248. Rodrigues' formolA. If A, /«, r denote the diraettaB
of the normal, and A* one of the principal radii of cunralora, th#
corresponding principal center of curvature will be given by the
formulae
(37) X = x^R\, l' = y + /f^, ^-s-f-^r.
As the point (x, y, «) describes a Una of enrvature
the normal section whose radius of eurvatitre is If, this
518 SURFACES [Xii,§249
curvature, as we have just seen, will describe a curve T tangent to
the normal MN ; hence we must have
dX_dY_dZ_
or, replacing X, Y, and Z by their values from (37) and omitting the
common term dR,
dx-\- Rd\ _ dy -\- Rdii _ dz-\- Rdy
The value of any of these ratios is zero, for if we take them by
composition after multiplying each term of the first ratio by X, of
the second by fi, and of the third by v, we obtain another ratio
equal to any of the three ; but the denominator of the new ratio is
unity, while the numerator
\dx ■}- fxdy -\-vdz + R(\ d\ -\- fj, dfi -\- v dv)
is identically zero. This gives immediately the formulae of Olinde
Rodrigues :
(38) dx + Rd\ = 0, di/ + Rdfji = 0, dz-\-Rdv = 0,
which are very important in the theory of surfaces. It should be
noticed, however, that these formulae apply only to a displacement
of the point (x, y, z) along a line of curvature.
249. Lines of curvature in parameter form. If the equations of the
surface are given in terms of two parameters u and v in the form
(19), the equations of the normal are
X-x _ Y-y _ Z -z
A ~ B ^ C '
where A, B, and C are determined by the equations (21). The
necessary and sufficient condition that this line should describe a
developable surface is, by § 223,
(39)
dx dy dz
ABC
dA dB dC
= 0,
where x, y, z, A, B, and C are to be replaced by their expressions
in terms of the parameters u and v; hence this is the differential
equation of the lines of curvature.
Xll,ja«] LTXES OF ilTRVAii RB ^10
As an example let ub nnd lue uii« ul curYAUirc ua iIm Mlsali
t saaretao^.
s
whose equation is equiyalent to the systeia
z = pcostf, yspsiaf. i e «#.
In this example the equations (or A, B, and C aie
-<lcoetf+Biln# = 0, - Aprini -^ Bpemi ^ V*9
Taking C = p, we And ^ = adn «, Jl = - a oosi^. Afier cxpMMtea
iication the differential equation (80) becomes
dp*-(p« + d^*« = or d0 = ±-^ ,
Choosing the sign +, for example, and IniegraUnf, w fhid
p + V^To* = ««•-•., or p^^l^-S-rHf-M],
The projections of these lines of cunrature on the sy piaae are aB
are easily constructed.
The same method enables us to form the • (if thr tvaad
degree for the principal radii of curvature. \'- an;.- ^v r: ]..:•
Af B, Cf Xj fi, V we shall have, except for ligti.
/* =
Va* + B*+C* Vi4M^B*+C«
We shall -adopt as the positive direction of the Dormal that vhidi
is given by the preceding equations. If i? it a phadpal radiot of
curvature, taken with its proper sign, the oo6ttliiiiUt of the eoir^
sponding center of curvature are
X^x-^-pA, Y^y-^pB, Z-«+^r,
where
R^py/A^-^^-k-C;
If the point (x, y, t) describes the line of car?alni« Uuif«it to lU
principal normal section whose radius of oarraturs b #, w« Iuitv
seen that the point (.Y, K, Z) describes a curve T whtefa is t S Bf s a l
t'> tlu' i..)rni:il to the surface. Heooo we must have
A B
620
SURFACES
[XII, § 260
or, denoting the common values of these ratios by dp + A",
rdx -\- pdA — AK = 0,
(40) \dy-^pdB-BK=0,
[dz + pdC - CK = 0.
Eliminating p and K from these three equations, we find again the
differential equation (39) of the lines of curvature. But if we
replace dx, dy, dz, dA, dB, and dC by the expressions
dx ^ dx ^
dC ^ dC ,
w-du + ^-dv,
du dv
respectively, and then eliminate du, dv, and K, we find an equation
for the determination of p :
(41)
dx dA
du " du
dx dA
dv^f" dv
dy ^ dB
du ^ du
dy ^ dB
dv '^ cv
dz dC
du ^ du
dz ^ dC
dv^f'dv
= 0.
If we replace p by r/ ^A'^ + 5^ -}- C^, this equation becomes an
equation for the principal radii of curvature.
The equations (39) and (41) enable us to answer many questions
which we have already considered. For example, the necessary
and sufficient condition that a point of a surface should be a para-
bolic point is that the coefficient of p^ in (41) should vanish. In
order that a point be an umbilic, the equation (39) must be satisfied
for all values of du and dv
As an example let us find the principal radii of curvature of the rectilinear
helicoid. With a slight modification of the notation used above, we shall have
in this example
x = Mcosu, y= usinu, z = av,
A = aeinv, B = — acosv, C = u,
and the equation (41) becomes
a2p2 _ ^2 _,. ^2^
whence E = ± (a^ + u'^)/a. Hence the principal radii of curvature of the helicoid
are numerically equal and opposite in sign.
250. Joachimsthal's theorem. The lines of curvature on certain
surfaces may be found by geometrical considerations. For example,
it is quite evident that the lines of curvature on a surface of revolu-
tion are the meridians and the parallels of the surface, for each of
XII, §251] LlNh.^ Ut tLttVATLUK Stl
these curves is tangent at erery point to OM of Hm ai« of iW
indicatrix at that point This if again ooDfinMd bj IIm
that the normals along a meridian form a plmo^ and tiM
along a parallel form a oiroular oone, — in eaoh oaM tho
form a developable surfaoe.
On a developable surface the first system of ham of
consists of the generators. The second syttem *»'*Ttfttff of Ike
orthogonal trajectories of the generators, that it, of tlM inTnlrtw of
the edge of regression (f 231). Theae oan be found bjr a tlifle qitd*
rature. If we know one of them, all the reit oan be found vitbom
even one quadrature. All of these resolta are eatilj Torifiod direelly.
The study of the theory of evolutes of a skew onnre lod Je^
chimsthal to a very important theorem, which is ofUn need in tknft
theory. Let S and s' be two surfaoes whoee line of intarteetkn C
is a line of curvature on each surface. Tho normal MN to 8 aloiy
C describes a developable surface, and the normal MS* to ^
C des(Tibes another developable surface. But each of
is normal to C. It follows from f 231 that \f two mrfmtm kmtft m
common line of curvature^ they inUrteet at a wmttmmt mm§U mUm$
that line.
Conversely, if tiro turfaew ifiUnmft at a esnitanf — ffi^ mmd \f
their line of intersection i» a line of e ur va iur $ am am§ ^ f As i , ii if
also a line of curvature on the other. For we hare aean tbat if one
family of normals to a skew curve C form a doTelopable aurfaee^
the family of normals obtained by turning eaoh of thn finft fa^Hy
through the same angle in its normal plane also form a devftlofiahU
surface.
Any curve whatever on a plane or on a sphere is a lino of cnra^
ture on that surface. It follows as a cocoUary to J<
theorem that the necessary and s^ffieient eam diii am thai •
or a spherical curve on any susfaee ekamid U a Um ^ mmrnima <t
that the plane or the sphere en whiek tMe ettr^e Um ekemid mi tka
surface at a constant -angle,
251. Dupln'8 theorem. We have already oontidvid [If 4S, 146]
triply orthogonal systems of surfaces. The origin of the tksaiy ol
such systems lay in a noted theorem due to Dopin, whieb we sknil
proceed to prove :
Given any three families qf surfaces wkiek fsrm a tnpiy erikefsmmt
system : the intersection <^any tw mgfmm ^ d}§m mt^mikm i§ •
line of curvature on each of (Aem.
522 SURFACES [XII, §251
We shall base the proof on the following remark. Let F(x^ y,z)=0
be the equation of a surface tangent to the xi/ plane at the origin. Then
we shall have, f or a; = y = « = 0, dF/dx = 0, dFjcy = 0, but dFjdz does
not vanish, in general, except when the origin is a singular point
It follows that the necessary and sufficient condition that the x and
y axes should be the axes of the indicatrix is that s = 0. But the
value of this second derivative s = d'^z/dx dy is given by the equation
d^F , d^F , d'^F , d^F , dF
dx dy dx dz dy dz dz^ dz
Since p and q both vanish at the origin, the necessary and sufficient
condition that s should vanish there is that we should have
d'^F _
^ ' dxdy
Now let the three families of the triply orthogonal system be given
by the equations
Fii^i Vi ^) = pi, Fi(p^, y, «) = />2, Fs{x, y, «) = p8,
where Fi, F2, F^ satisfy the relation
^ ^ dx dx dy dy dz dz
and two other similar relations obtained by cyclic permutation of
the subscripts 1, 2, 3. Through any point AT in space there passes,
in general, one surface of each of the three families. The tangents to
the three curves of intersection of these three surfaces form a trirec-
tangular trihedron. In order to prove Dupin's theorem, it will be
sufficient to show that each of these tangents coincides with one of
the axes of the indicatrix on each of the surfaces to which it is
tangent.
In order to show this, let us take the point M as origin and the
edges of the trirectangular trihedron as the axes of coordinates;
then the three surfaces pass through the origin tangent, respec-
tively, to the three coordinate planes. At the origin we shall have,
for example,
(S).=«. ©.=»■ ©'<».
(ti'». ©).=»• (fii-»-
©.=«■ f^i«»' ©=»•
XlLjaoiJ LINKS OF CURVATf'RB
The axes of ob and y will be the aut ol \Xm iodtelrii of tW
^(a^» y» ») = at the origin if {i^F^J^ by). • a To
is the oaae, let ut differentiate (43) with raspeoi to y,
terms which vaniah at the origin ; we find
or
(44)
(SI m."
From the two relations analogous to (43) we oouUl dedoM two
equations analogous to (44), which may be written down hj cjdie
permutation :
From (44) and (45) it is evident that we shall have alao
(fgi=«. (ai-' m-''
which proves the theorem.
A remarkable example of a triply orthogonal syaleai it funiaked
by the confocal quadrics diseussed in § 147. It wm dnntHleM llw
investigation of this particular system which led Dopia to IIm goh
eral theorem. It follows that the lines of cunratnro on an ellipsoid
or an hyperboloid (which had been detemuBMl ynwiomiy by Mo^ge)
are the lines of intersection of thai tnrCnae with ili
The paraboloids represented by thn
y-X
-2«- A,
where A is a variable parameter, form ino li Mf tAyiy
system, which determines the lines of eoimtart es IIm
Finally, the system discussed in | 346,
is triply orthogonal
524 SURFACES [XII, §252
The study of triply orthogonal systems is one of the most interest-
ing and one of the most difficult problems of differential geometry.
A very large number of memoirs have been published on the subject,
the results of which have been collected by Darboux in a recent
work.* Any surface ^ belongs to an infinite number of triply
orthogonal systems. One of these consists of the family of surfaces
parallel to S and the two families of developables formed by the
normals along the lines of curvature on S. For, let be any point
on the normal MN to the surface S at the point M, and let MT
and MT' be the tangents to the two lines of curvature C and C
which pass through AI; then the tangent plane to the parallel sur-
face through O is parallel to the tangent plane to S at M, and the
tangent planes to the two developables described by the normals to
S along C and C are the planes MNT and MNT', respectively. These
three planes are perpendicular by pairs, which shows that the system
is triply orthogonal.
An infinite number of triply orthogonal systems can be derived
from any ouq known triply orthogonal system by means of succes-
sive inversions, since any inversion leaves all angles unchanged.
Since any surface whatever is a member of some triply orthogonal
system, as we have just seen, it follows that an inversion carries the
lines of curvature on any surface over into the lines of curvature on
the transformed surface. It is easy to verify this fact directly.
252. Applications to certain classes of surfaces. A large number of problems
have been discussed in which it is required to find all the surfaces whose lines
of curvature have a preassigned geometrical property. We shall proceed to
indicate some of the simpler results.
First let us determine all those surfaces for which one system of lines of
curvature are circles. By Joachimsthal's theorem, the plane of each of the
circles must cut the surface at a constant angle. Hence all the normals to the
surface along any circle C of the system must meet the axis of the circle, i.e.
the perpendicular to its plane at its center, at the same point 0. The sphere
through C about as center is tangent to the surface all along C ; hence the
required surface must be the envelope of a one-parameter family of spheres.
Conversely, any surface which is the envelope of a one-parameter family of
spheres is a solution of the problem, for the characteristic curves, which are
circles, evidently form one system of lines of curvature.
Surfaces of revolution evidently belong to the preceding class. Another
interesting particular case is the so-called tubular surface^ which is the envelope
of a sphere of constant radius whose center describes an arbitrary curve r. The
characteristic curves are the circles of radius R whose centers lie on V and
whose planes are normal to r. The normals to the surface are also normal to r ;
* Le<;on8 aur les ayatemea orthogonauz et lea coordonn^ea curvilignea, 1898.
Xn,§282] LINES OF LLKVATLKE
hence the eeoond lygtem of Uaei of eomtnre aro tko ttntt Is
is cat by the developable ■arfMte which nwy be fbratd tnm ilMMfVitoier
If both lystema of Udcs of cart»tare on » surfaoe v dirla*, K k diw to^
the preceding argument that the earfaoe may be thooghi of m ike eafiloM «f
either of two one-parameter famiUea of ■pheres. LiClH* ^. ^ be «w ^m
spheres of the first family, C|, Ct, C| the oorreapoodlag ^^ra ri artilii) im9iL
and Mi,Mt,Mtthe three poinu in whieh Ci, Ct, Ct ai«e«t byalteaelcwf^
ture C of the other system. The ipbere 8' which ia t^«i fm
C is also tangent to the spheres 5|, ^, fis at Jf, , if,, Jjf,,
the required tur/aee is the env€U>pe qf a /aimUif qf ijily mck ^ «4kA
three fixed spheres. This surface is the well.jEDOWB D^te cydMs.
gave an elegant proof that any Dupin cydide la the mfaee into mhkk * <
anchor ring is transformed by a certain invwiioii. Let y
is orthogonal to each of the three fixed q>h«rai tfi, Ai, 8%, Aa kt\
pole is a point on the circumference of y earriee tliat etrelt iMo a
OCy, and carries the three spheres 0i, 8g, 8t into three ipharai 2tt Se* Zs
orthogonal to OCT, that is, the centers of the tiauComMd iphewi lia «• OCT.
Let Ci, C^, C^ be the intersecUons of these Wfimm wflk a«y ptaM miniil
OO', C a circle tangent to each of the dreles C(. C«, C|, tad S* Ite iplMw
on which C is a great circle. It is clear that If renaina tafit to aMb of Ika
spheres Zi, 2^, Zg as the whole figure is rerolTed aboot 0<X, aad that the
envelope of 2' is an anchor ring whose meridlaB la the eireie C
Let us now determine the surface for which all of the liaea of emnuan of
one system are plane curves whoee pUuiea are all parallal Let aa laha tka ^
plane parallel to the planes in which theaa Unea of oorvalua lla. aad let
zcosar + ysinas Fia^i)
be the tangential equation of the eeetioB of the aorfaoe by a panaH to the ay
plane, where F(a, z) is a function of a and s which dapaoda «
under consideration. The coordinates z and y of a point of
given by the preceding equation together with the ai|Ballos
dF
-zaina + ycoaas -—•
9a
The formulsB for z, y, z are
(46) z = Fcoear sina, y = Faintf + —ooea. i « s.
Any surface may be represented by eqoatloaa of
function F{a, z) properly. The only a ioa p tiflM are tfca ^^^^
directing plane is the xy pUne. It is aaay to abow that tho oaaAolaMa A^B^C
of the tangent plane may be taken to be
il = coaa, Bsdna, C^ ^
hence the cosine of the angle between the nofaal and the t aiia la
-F.(a.t)
" Vl+ j;(a,a)
In order that all the aeotiona by plaaea parmllal to Iha i
ture, it is necessary and aoflkient, by
626 SURFACES [XII, §263
these planes cut the surface at a constant angle, i.e. that v be independent of a.
This is equivalent to saying that Fz{a^ z) is independent of a, i.e. that F{a^ z)
is of the form
F{a,z) = <p{z) + rl^{a),
where the functions </> and yf/ are arbitrary. Substituting this value in (46), we
see that the most general solution of the problem is given by the equations
X = yl/{a) cos a — r/{cc) sin a + <f>{z) cos a ,
(47) -[ 3/ = ^(«) sin a + ^'(a) cos a: + <f>{z) sin a ,
fx = \f/{a) cos a — yf/{
-j 3/ = ^(«) sin a + ^'(
These surfaces may be generated as follows. The first two of equations (47),
for z constant and a variable, represent a family of parallel curves which are
the projections on the xy plane of the sections of the surface by planes parallel
to the xy plane. But these curves are all parallel to the curve obtained by set-
ting <p{z) = 0. Hence the surfaces may be generated as follows : Taking in the
xy plane any curve whatever and its parallel curves^ lift each of the curves verti-
cally a distance given by some arbitrary law ; the curves in their new positions form
a surface which is the most general solution of the problem.
It is easy to see that the preceding construction may be replaced by the
following : The required surfaces are those described by any plane curve whose
plane rolls without slipping on a cylinder of any base. By analogy with plane
curves, these surfaces may be called rolled surfaces or roulettes. This fact may
be verified by examining the plane curves a = const. The two families of lines
of curvature are the plane curves z = const, and a = const.
IV. FAMILIES OF STRAIGHT LINES
The equations of a straight line in space contain four variable
parameters. Hence we may consider one-, two-, or three-parameter
families of straight lines, according to the number of given relations
between the four parameters. A one-parameter family of straight
lines form a ruled surface. A two-parameter family of straight
lines is called a line congruence, and, finally, a three-parameter
family of straight lines is called a line complex.
253. Ruled surfaces. Let the equations of a one-parameter family
of straight lines (G)he given in the form
(48) x = az+pj y=:bz-{-qy
where a, ft, jo, q are functions of a single variable parameter u. Let
us consider the variation in the position of the tangent plane to the
surface S formed by these lines as the point of tangency moves along
any one of the generators G. The equations (48), together with the
equation z = z, give the cobrdinates x, y, zot & point M on S in terms
Xll.fgtt] FAMILIES OF STRAIGHT LUiM^
of the two parametert t ud u ; h«iio«, bj f M^ tb* •^mMm of tk§
tftDgent plane at Af U
X-x r-y Z-«
-0,
where a\ 6', />', q' denote the deriTatiTM of m^h,p,f vtUi
to u. Replacing x and y by <» + /> and M + f ,
simplifying, this equation beoomet
(49) (b'z + q')(X - aZ ^p) - (a'« +/»')(K - *Z - f) - 0.
In the first place, We see that this plana always passw tliroofk tlM
generator G^ which was erident a priarif and moreorer, tlial Um plaA«
turns around G as the point of tangenoy At mores aloof G, ai WmI
unless the ratio (a'x + p')/{h'x + g') is indepsodoDt of t, i^. uiteia
a';' — d'^' = 0, — we shall discard this special ease in what folkmt.
Since the preceding ratio is linear in s, eTOty plane throofh a pm*
erator is tangent to the surface at one and only ooa point As tbt
point of tangenoy recedes indefinitely along the gao«ralor in oHbar
direction the tangent plane P approaches a limiting poatUoft /**,
which we shall call the tangent plane at the poini ai it\^miif OB tlMI
generator. The equation of this limiting piano P* is
(60) b\X - oZ - /») - a'(r - *Z - f ) - 0.
Let o> be the angle between this plane P* and the tanftnt plaao P al
a point M (x, y, z) of the generator. The diroetioo ootiiMS («', /T, /^
and (a, ft y) of the normals to P' and /* are proportional lo
b\ -a\ o'4-aA'
and
*'« + ?', -(a'«4-p'), 6(«'»+/>')-«i,<'«^?').
respectiyely ; hence
cos « = aa- + /J/r + ry' - ;^v5i== •
where
After an easy reduction, we find, by Lagrange's identity (1 15tV
(51) tan« = ^j;^:p^-^-«^ ^^
528 SURFACES [XII, §253
It follows that the limiting plane P' is perpendicular to the tangent
plane Pi at a point Oi of the generator whose ordinate Zi is given by
the formula
xroN ^ _ ay + b'q' + (ab' - ba^jaq' - bp')
^^ ' A a'^ + b''' + {ab'-ba'y
The point Oi is called the central point of the generator, and the tan-
gent plane Pj at Oi is called the central plane. The angle $ between
the tangent plane P at any point M of the generator and this central
plane Pi is 7r/2 — <o, and the formula (51) may be replaced by the
formula
^,. ^ - _l (l:ziil - r^" + b'^ + (g^- - ba^Mz - gi2,
V^ C - ^2 (^f^r _ jr^f) Vl + a2 -f ^2
Let p be the distance between the central point Oi and the point My
taken with the sign + or the sign — according as the angle which
OiM makes with the positive z axis is acute or obtuse. Then we
shall have p = (z — z^) Vl -f a* + b^, and the preceding formula may
be written in the form
(53) tan^ = A:p,
where k, which is called the parameter of distribution^ is defined by
the equation
_ a^'^^h^^ + jab^ -bay
^ ^ ~ (a'^' - b^p%l + a=^ 4- b^)
The formula (53) expresses in very simple form the manner in which
the tangent plane turns about the generator. It contains no quantity
which does not have a geometrical meaning : we shall see presently
that k may be defined geometrically. However, there remains a cer-
tain ambiguity in the formula (53), for it is not immediately evident
in which sense the angle 6 should be counted. In other words, it is
not clear, a priori^ in which direction the tangent plane turns around
the generator as the point moves along the generator. The sense of
this rotation may be determined by the sign of k.
In order to see the matter clearly, imagine an observer lying on a
generator G. As the point of tangency M moves from his feet toward
his head he will see the tangent plane P turn either from his left
to his right or vice versa. A little reflection will show that the
sense of rotation defined in this way remains unchanged if the
observer turns around so that his head and feet change places.
Two hyperbolic paraboloids having a generator in common and
XU.f2B3j FAMILIES OF STRAIGHT LUftt
lying synunetrically with ratpaet to a plaa« tfi t^^ff fc g^
give a clear idea of the two pottible tiliuilkNM. iMm _
the axes in such a way that the new origin is at the fninl poiat #«,,
the new x axis is the generator (* itself, and the cs pbftt It tW mm-
tral plaiie I\. It is evident that the value of th« piriMStw oC 4k^
tribution (54) remains unchanged during this moveaMitol th« nm,
and that the formuhi (53) takes the form
(53') tantf = A«,
where $ denotes the angle between the jr« pUne I\ and th« uagrai
plane /', counted in a convenient sense. For the value of «« which
corresponds to the z axis we must have aw^bwrnparnqmO^ and th«
equation of the tangent plane at any point Mi of tiial m^
(6'«4-O'^-(a'«+p')r-0.
In order that the origin be the central point % n d the
central plane, we must have also a' as 0, 7' ss ; henee Um
of the tangent plane reduces to F = (^**/p')X, and the formola (M)
gives k=- b'/p\ It follows that the angle $ in (58*) abodd bo
counted positive in the sense from Oy toward Osl If Ifao oiiMili
tion of the axes is that adopted in f 228, an obeerror Ijinf in tiM
X axis will see the tangent plane turn from his left towaid Iba rifltt
if k is positive, or from his right toward his left if ir it ncgnliTO.
The locus of the central points of the generators of a nUed torlbet
is called the line of striction. The equations of this eiirro in ItfM
of the parameter u are precisely the equations (48) and (51>.
Note. If a'y' = b'p' for a generator (i, the tangent plane is tiM
same at any point of that generator. If this relation it tatiaiii
for every generator, i.e. for all values of ti, the ruled anrfiMt It a
developable surface (§ 223), and the results preriouslj obtaintd cm
be easily verified. For if fl' and 6' do not vanish tlBnltaBOOnily*
the tangent plane is the same at all points of any gsaviioff G,
and becomes indeterminate for the point a — -!>*/•*«— f'/*'f ta-
for the point where the generator toittlitt ita tovalopa. It it tasgr
to show that this value for x is the sane as that flvw Iqr (W) wbm
a'q' = b'p'. It follows that the line of strielkm hiooatt Ika adga
of regression on a developable surface. The paranaltr of (
is infinite for a developable.
If a' = 6' = for eveiy generator, tha tarfa^ b a eyUadar
the central point is indeterminate.
630 SURFACES [XII, § 254
254. Direct definition of the parameter of distribution. The central
point and the parameter of distribution may be defined in an entirely
different manner. Let G and Gi be two neighboring generators cor-
responding to the values u and u ■}- hoi the parameter, respectively,
and let G^ be given by the equations
(55) ic = (a 4- Aa) z -{- p + A;?, y = (b -\. A6) « + 2' + Ag'.
Let 8 be the shortest distance between the two lines G and G^ , a the
angle between G and G^y and (Z, F, Z) the point where Cr meets the
common perpendicular. Then, by well-known formulae of Analytic
Geometry, we shall have
_ Aa Ay 4- A5 A/? + (o^ A& — ^ Aa)[(a + Aa)Aq — (b + Ab)Ap']
(Aa)2 + (Aby -\-(aAb-b Aay *
8 =
Act Ag^ — A6 Aj9
sm a =
V(Aa)2 + {Abf -\-(aAb-b Aaf
-s/{Aay + (A^)'' -{-{aAb-b Aaf
-si a? + ^>2 +1 V(a + Aa)2 + (6 + A^»)2 + 1
As h approaches zero, Z approaches the quantity z^ defined by (52),
and (sin «)/8 approaches k. Hence the central point is the limiting
position of the foot of the common perpendicular to G and G^j, while
the parameter of distribution is the limit of the ratio (sin a)/h.
In the expression for 8 let us replace Aa, A5, Ajo, Ag- by their
expansions in powers of h\
h^
Aa = /la' + :r-^ a" H
and the similar expansions for A&, Ap, Aq. Then the numerator of
the expression for 8 becomes
AaAq-Ab Ap = h\a'q' - b'p') + j (a"q' -f a'q" - b"p'-b'p") -f • • .,
while the denominator is always of the first order with respect
to h. It is evident that S is in general an infinitesimal of the first
order with respect to h, except for developable surfaces, for which
a'q' = b'p'. But the coefficient of h^/2 is the derivative of a'q' — b'p'-^
hence this coefficient also vanishes for a developable, and the shortest
distance between two neighboring generators is of the third order
(§ 230). This remark is due to Bouquet, who also showed that if
this distance is constantly of the fourth order, it must be precisely
zero; that is, that in that case the given straight lines are the
XII,»aB6] FAMILIES OP STRAIGHT UKE8 5tt
tangents to a plane ovrre or to a oaoiccl nthm, la otdir lo pt«i«
this, it is sufficient to cany the derelopment of ^a^f - A^Ajito
terms of the fourth order.
255. Con^niencee. Focalsoxfaotof acoagivnfit. Bm, twti rarawi^
family of straight linee
(66) ««=a«+/», p^hm-i-q,
where a, 6, p, q depend on two parameten a and ^ ii cmlled a lfa#
congruence. Through any point in tpaoe tbeie paae, ia ftaeral, a
certain nimiber of lines of the oongnienoe, for the two eqo^loM (56)
determine a certain number of definite aete of valaee of a aad $ vhM
X, y, and z are given definite values. If any relatioo betwei a wni B
be assumed, the equations (56) will represent a ruled sttrfaoe, whidi
is not usually developable. In order that the turiaoe be
we must have
dadq — dhdpssQf
or, replacing da by (da /da) da + (da/dp) dfi, eta,
(57)
This is a quadratic equation in dfi/da. Solving it^ we iboaUl
ally obtain two distinct solutions,
(58) ^ = ^t(«'«' 5f-M*'»»
either of which definee a developable sorfaoe. Under vety
eral limitations, which we shall etaie preoiaely a Utile later
which we shall just now suppose fulfilled, eneh of
is satisfied by an infinite number of fuooiiooa of a, and eneli of
has one and only one solution which aasumee a given valne
a=: ao. It follows that eveiy straight line G oi
belongs to two developable sorfaoee, all of wboee
members of Uie congnienoe. Let T and T be the edfee of
of these two developables, and ^1 and J ' the poinCa where G
T and r', respectively. The two poinU A and A ' aio called Ike /Mel
points of the generator G. They may be fonnd at foUowa wilkoel
integrating the equation (57). The oidinale a of one of
must satisfy both of the equation!
532 SURFACES [XII, §255
or, replacing da, db, dp, dq by their developments,
Eliminating z between these two equations, we find again the equa-
tion (57). But if we eliminate d^/da we obtain an equation of the
second degree
whose two solutions are the values of z for the focal points.
The locus of the focal points A and A^ consists of two nappes
2 and S' of a surface whose equations are given in parameter form
by the formulae {50) and (59). These two nappes are not in general
two distinct surfaces, but constitute two portions of the same ana-
lytic surface. The whole surface is called the focal surface. It is
evident that the focal surface is also the locus of the edges of regres-
sion of the developable surfaces which can be formed from the lines
of the congruence. For by the very definition of the curve F the
tangent at any point a is a line of the congruence ; hence a is a
focal point for that line of the congruence. Every straight line
of the congruence is tangent to each of the nappes ^ and 2', for it
is tangent to each of two curves which lie on these two nappes,
respectively.
By an argument precisely similar to that of § 247 it is easy to
determine the tangent planes at A and A' to % and 2' (Fig. 51).
As the line G moves, remaining tangent to F, for example, it also
remains tangent to the surface 2'. Its point of tangency A^ will
describe a curve y' which is necessarily distinct from F'. Hence
the developable described by G during this motion is tangent to 2'
at A\ since the tangent planes to the two surfaces both contain the
line G and the tangent line to y'. It follows that the tangent plane
to 2' at .4' is precisely the osculating plane of F at ^. Likewise,
the tangent plane to ^ at A is the osculating plane of F' at A '.
These two planes are called the focal planes of the generator G.
It may happen that one of the nappes of the focal surface degen-
erates into a curve C. In that case the straight lines of the con-
gruence are all tangent to % and merely meet C. One of the
families of developables consists of the cones circumscribed about 2
XII. §258] FAMILIES OP STRAIGHT LINtt
whose vertioes are on C. If both of the nappes of iIm
degenerate into ourves C and C\ the two ^■■ftMff of
consist of the oones through one of the ourfoe wliim Toitjiwi lie
on the other. If both the cunret C and C* are lifaifitt Iteaa, tkm
congruence U called a /ifieor e w ^prweiiet.
256. Congmence of normala. The nonnali to aaj ttiriSMO eiM^llj
form a congruence, but the converse is not true: thart eiitls ao
surface, in general, which is normal to every line of a fives eos-
gruence. For, if we consider the congruenoe formed bjr tlie nnfili
to a given surface S^ the two nappes of the focal surface are evidesUjr
the two nappes 2 and 2' of the evolute of S (| 247), and we have son
that the two tangent planes at the points A and A ' where the turn
normal touches 2 and X stand at right angles. This is a ehaiacltr
istic property of a congruence of normals, as we shall see by Ujtaf
to find the condition that the straight line (56) should alwaja rsouua
normal to the surface. The necessary and sufficieoi o n od it ka UmI il
should is that there exist a function /(a, fi) such that the anrfMa S
represented by the ec[uations
(60) x = az+p, yssbt-^q, M^J{a,fi)
is normal to each of the lines (O). It follows that we most have
da da da
or, replacing x and y by a« + /» and As + y , respeetively, and divid-
ing by Va«-H^*-fl,
The necessary and sufficient condition that
patible is
(61)
(62)
634 SURFACES [XU. § 266
If this condition is satisfied, z can be found from (61) by a single
quadrature. The surfaces obtained in this way depend upon a con-
stant of integration and form a one-parameter family of parallel
surfaces.
In order to find the geometrical meaning of the condition (62), it
should be noticed that that condition, by its very nature, is inde-
pendent of the choice of axes and of the choice of the independent
variables. We may therefore choose the z axis as a line of the con-
gruence, and the parameters a and fi as the coordinates of the point
where a line of the congruence pierces the xy plane. Then we shall
have p = a,q = P, and a and b given functions of a and ^ which van-
ish for a = ^ = 0. It follows that the condition of integrability, for
the set of values a = ^ = 0, reduces to the equation da/dp ==■ db/da.
On the other hand, the equation (57) takes the form
which is the equation for determining the lines of intersection of
the xy plane with the developables of the congruence after a and
p have been replaced by x and y, respectively. The condition
da/ dp = db/da, for a = ^ = 0, means that the two curves of this
kind which pass through the origin intersect at right angles ; that
is, the tangent planes to the two developable surfaces of the congru-
ence which pass through the z axis stand at right angles. Since the
line taken as the z axis was any line of the congruence, we may state
the following important theorem:
The necessary and sufficient condition that the straight lines of a
given congruence be the normals of some surface is that the focal planes
through every line of the congruence should be perpendicular to each
other.
Note. If the parameters a and /S be chosen as the cosines of the angles which
the line makes with the x and y axes, respectively, we shall have
"^ ft= . ^ — Vl-f-a2 + 6»= ^
Vl - a2 _ |32 VI - a^ _ /32 Vl - aa - /3a
and the equations (61) become
(68)
ea\y/i-a^-(^J da Ha '
. a/3 \y/l -a'- fit/ »/s a^
XU.J257]
FA\fTTJF>i <«w w-raAIGIIT LUTtt
Then the condiiion of tutegrmhUitj (08) fodaevio Itoloni «f /«« • W|^
meana that p and « inuM bo tiM pMtkl 4irtvMl««oliW MM foMiktt #h^ii|,
«-
»F
where F{a, ^) can be foond by a eingle quadrmuuv. It
solution of the total differential equation
i^M • ■ «•
d
whence
(7KS^^)-(-^r*'^)-(-S*'»')'
where C ia an arbitrary constant.
857. Theorem of Malua. If rayi of light from a pdat i
refracted) by any surface, the reflected (or refraeled) rays ai« dw
each of a family of t>arallel surfaoea. Thie theorem* which U tfw to I
been extended by Canchy, Dupin, Oerfoiuie, and Qoalelet to Uto CMO ct aaj
uuiuber of socoeeaive reflectiona or refraotiooa, and wo may eloio Hw
more general theorem :
// a family of rays of lighi ore normai to mmM mtr/on at
that property after any number ttf rqflaeKoiia oi
oay ua»<, uwy
Since a reflection may be regarded aa a rofraetlon of lades - 1« llli tvltallf
sufficient to prove the theorem for a single refraction. Let bo a
inal to the un refracted rays, mU an incident ray which moots Iks
separation Z at a point AT, and UU the refracted r^. By
incident ray, the refracted ray, and the nomal UV lie in a
angles t and r (Fig. 52) satisfy the relation
n sin i — sin r. For definiteness wt
pose, as in the figure, that n is
unity. Let I denote the diatanoo i^N^ and
let us lay off on the refracted ray ezteDded
a length V = Mm' equal to k times 2, whers
A; is a constant factor which we shall detsr-
mine presently. The point m' descrHMO a
surface S'. We shall proceed to show that
k may be chosen in such a way that Jfm' Is
normal to S'. Let C be any curre on fl.
As the point m describes C the point U
describes a curve r on the snrteoo Z, and
the corresponding point m' describee anotlior
curve C on S'. Let •, <r, t' be the lengths of ths
C measured from correspondla^
w the angle which the tongont UT\ to F owkeo vllk Ifeo taafnt UT
normal secUon by tho nomal pUno throngh tko tanllHl my, m^ ♦ Md
angles which ATT, makes with ifm and Jfm',
cos 0, for example, let us lay off on ifm a unit
of llMtlH<OS
cr.
536 SURFACES [XII, §258
first directly, then by projecting it upon MT and from MT upon MTi. This,
and the similar projection from Mm^ upon MTi, give the equations
cos <p = sin i cos w , cos <f>^ = sin r cos w .
Applying the formula (10') of § 82 for the differential of a segment to the seg-
ments Mm and Mm% we find
dl = — da cos w sin i ,
dl' = — d<T cos w sin r — (f s' cos ,
where ^ denotes the angle between m'Jf and the tangent to C\ Hence, replacing
dl' by k di, we find
cos w d(r{k sin i — sin r) = ds" cos ^ ,
or, assuming k = n,
ds' cose = 0.
It follows that Mm^ is normal to C, and, since C is any curve whatever on
S\ Mm/ is also normal to the surface S'. This surface -S' is called the anti-
caustic surface, or the secondary caustic. It is clear that S' is the envelope of
the spheres described about M as center with a radius equal to n times Mm ;
hence we may state the following theorem :
Let us consider the surface S which is normal to the incident rays as the envelope
of a family of spheres whose centers lie on the surface of separation 2. Then the
anticaustic for the refracted rays is the envelope of a family of spheres with the
same centers, whose radii are to the radii of the corresponding spheres of the first
family as unity is to the index of refraction.
This envelope is composed of two nappes which correspond, respectively,
to indices of refraction which are numerically equal and opposite in sign. In
general these two nappes are portions of the same inseparable analytic surface.
258. Complexes. A line complex consists of all the lines of a three-parameter
family. Let the equations of a line be given in the form
(64) x = az +Py y = bz -\- q .
Any line complex may be defined by means of a relation between a, 6, p, q of
the form
(65) F(a,6,p,g) = 0,
and conversely. If F is a polynomial in a, 6, p, g, the complex is called an
alge^aic complex. The lines of the complex through any point (xo , yo , Zo) form
a cone whose vertex is at that point ; its equation may be found by eliminating
a, 6, p, q between the equations (64), (65), and
(66) Xo = azo + P, yo = ^Zo + q.
Hence the equation of this cone of the complex is
(67)
,/ x - xq y -yo xqz- xzq ypz - yzo \ _
\z - Zo ' z - zo ' z - Zo ' z - Zo I
Similarly, there are in any plane in space an infinite number of lines of the
complex ; these lines envelop a curve which is called a curve of the complex.
If the complex is algebraic, the order of the cone of the complex is the same as the
XII. §2M] FAMTITF>i nF >;THiU«HT L1NG8 MJ
claat of the curve of " f t^tm ^
the complex whifli _. ^
of tangeuu which c»a be dra ^^
in the plane P. As the nnmlK : „ „ i_ ., .
proved.
If the cone of the complex b alwAyt a pUne, the eui « fijiiv
and the equation (06) in of the form
(68) Aa-^Bb + Cp-^Dq-k- E{aq - Af») + Fs 0.
Then the locus of all the lines of the complex throagh aoy glvea potei (■», f^, s^
is the plane whone equation is
(69^ i ^^'^ - **> "^ ^<^ " "'•^ + ^^'•' " *•*>
The curve of the complex, since it must be of elssi unity,
point, tliat is, all the lines of the complex which lie In s
single point of that plane, which is called the pole or the/ecML A
plex therefore establishes a correspondence between the points md tkt pltMi
of space, such that any point in space co r w ep onds to a plane throofb that poim,
and any plane to a point in that plane. A conrwpondencii is also eslaUWMd
among the straight lines in space. Let D be a stialght line which doai not
belong to the complex, F and F* the foci of any two ptinsi thronfh A nn^ A
the line FF\ Every plane through A has Its focus at Its point of iMsnssllsn #
with the line D, since each of the lines ^F and fF" evfclMUly
complex. It follows that every line which meets both D and A
complex, and, fnially, that the focmi of any plane tkronfh D Is the
that plane meets A. The lines D and A are called CM^^pAttnse; each of i
is the locus of the foci of all planes through the other.
If the line IJ recedes to infinity, the pUuies throagh It
it is clear that the foci of a set of parallel planes lie on a atralght llno^
always exists a plane such that the locus of the fod of the planes pnrmM Is II
is perpendicular to that plane. If this particular Une be tahan as the s ails,
the plane whose focus is any point on the i axis Is painllsl to the sy plana. Wf
(60) the necessary and sufficient condition that this shonid be the OMi li IhiA
^ = £ = C = D = 0, and the equaUon of the oompla tahas the sla^ 9mm
(70) ofl - 6p + JT = 0.
The plane whose focus is at the point (x. y, s) Is flt»n by the mm0tmk
(71) Xy-r« + ir(Z-i)«o.
where X, F, Z are the running coitrdlnalss.
As an example let us determine the cunres whoa* i«o|t<^t.i* i^^^ig "*
preceding complex. Given such a cunrr, whooe eoQrdlimieB », y. t aw
functions of a variable parameter, the eqnatioaa of Iho IMpM it anf pi
if *t *t
538 SURFACES [XII, Exs.
The necessary and sufficient condition that this line should belong to the given
complex is that it should lie in the plane (71) whose focus is the point (x, y, z),
that is, that we should have
(72) xdy — ydx = Kdz.
We saw in § 218 how to find all possible sets of functions x, y, z of a single
parameter which satisfy such a relation ; hence we are in a position to find
the required curves.
The results of § 218 may be stated in the language of line complexes. For
example, differentiating the* equation (72) we find
(73) xd^y-yd2x = Kd^z,
and the equations (72) and (73) show that the osculating plane at the point
(x, y, z) is precisely the tangent plane (71) ; hence we may state the following
theorem :
If all the tangents to a skew curve belong to a linear line complex, the osculating
plane at any point of thai curve is the plane whose focus is at that point.
(Appell.)
Suppose that we wished to draw the osculating planes from any point O in
space to a skew curve r whose tangents all belong to a linear line complex. Let
M be the point of contact of one of these planes. By Appell's theorem, the
straight line MO belongs to the complex ; hence M lies in the plane whose focus
is the point 0. Conversely, if the point M of r lies in that plane, the straight
line MO, which belongs to the complex, lies in the osculating plane at Jf ; hence
that osculating plane passes through 0. It follows that the required points are
the intersections of the curve with the plane whose focus is the point (see
§218).
Linear line complexes occur in many geometrical and mechanical applica-
tions. The reader is referred, for example, to the theses of Appell and Picard.*
EXERCISES
1. Find the lines of curvature of the developable surface which is the
envelope of the family of plan s defined in rectangular coordinates by the
equation
z = ax + y</>{a) + RVTV^T^^o),
where or is a variable parameter, 0(a) an arbitrary function of that parameter,
and R a given constant.
[Licence, Paris, August, 1871.]
2. Find the conditions that the lines x = az + «, y = &2 + /3, where a, b, a, /S
are functions of a variable parameter, should form a developable surface for
which all of the system of lines of curvature perpendicular to the generators lie
on a system of concentric spheres.
[Licence, Paris, July, 1872.]
* Annates acientiflquea de I'Ecole Normale sup^rieure, 1876 and 1877.
XII, Ex«.] - rg
3. Detarmine Um Uats of eorrmtiuv of Um
i&DguUr co6rdiiuktM it
^«0Q8S00i|f.
^mim,Mif,im%.l
4. CoDiidcr Um eUljMold of Hum
X* ^ ^
and the ellipUcal section K in the xm plftiie. Plod, «l mtct potBi M oi g
values of the principal radii of curvature H| and A| of iIm ''^ ff r ^ , tk ac-
tion between Iti and Ki, 8) the loci of the coBlon of ewilaiv ol Iko
•ectioDs as the point M deacribea the ellipot E.
5. Derive the e<|uation of the
ture at any point of iLc paraboloid defined by the
a 6
Also express, in terms of the Tariablo s^ wdi of tbo
ture at any point un the line of interMctkm of Um
paraboloid defined by the equaUon
-^ + ^^ = ,._X.
a-X 6-X
[liesMc, Fjuk,
6. Find the loci of the centers of curvature of Ite ptt
paraboloid defined by the equation qr = as as the potel of tko
the X axis,
7. Find the equation of the aorfaco whleh It Um Ioom ol Ikt
vature of all the plane sections of a given enrfaM S bv
through the same point M of the mutacr
8. Let Jf r be any tangent line at a point JT of a I
center of curvature of the section of the sorfaM by any
and (y the center of curvature of the evolote of
locus of (/ as the secant plane revolvet aboot MT.
[Lkmtt, CTwMl , My. IM.]
9. Fhid the asymptotic linet on tht anchor rii«
about one of iu tangents. ^ ^
10. Let C be a given conre In the « plant in a tfm&m ol raaiMirinr taMil
nates. A surface is described by a drcio wkoat pin
gy plane and wfaota ea1» itrihst Uie conre C, wMit tfct
a vray that tho eiido alwaya HMtla tte s axis. Dwivo
of the asymptoUc linet on lUi surtMt, Inking at Um
640 SURFACES [XII, Exs.
coordinate z of any point, and the angle Q which the radius of the circle through
the point makes with the trace of the plane of the circle on the xz plane.
Apply the result to the particular case where the curve C is a parabola
whose vertex is at the origin and whose axis is the x axis.
[Licence^ Paris, July, 1880.]
11. Determine the asymptotic lines on a ruled surface which is tangent to
another ruled surface at every point of a generator A of the second surface,
every generator of the first surface meeting A at some point.
12. Determine the curves on a rectilinear helicoid whose osculating plane
always contains the normal to the surface.
[Licence, Paris, July, 1876.]
.18. Find the asymptotic lines on the ruled surface defined by the equations
x = (l + u)cosv, y = (1 — w)sint), z = u.
[Licence, Nancy, November, 1900.]
14*. The sections of a surface S by planes through a straight line A and the
curves of contact of the cones circumscribed about S with their vertices on A
form a conjugate network on the surface.
[KOENIGS.]
15*. As a rigid straight line moves in such a way that three fixed points
upon it always remain in three mutually perpendicular planes, the straight line
always remains normal to a family of parallel surfaces. One of the family of
surfaces is the locus of the middle point of the segment of the given line bounded
by the point where the line meets one of the coordinate planes and by th